Longyan - Divine Fox

If you don't have Divine Fox yet but want to try her out, she is available as a Trial Operative the Events menu. (Events -> Operative Trial -> Divine Fox)

Longyan offers both strong single-target damage and strong AoE-damage, but this comes at the cost of her benefitting a lot from certain support effects (Aptitudes) which severely limits her options for teambuilding, and Longyan also has mobility so bad it can only be described as tragic.
Longyan also has a lot of effects proc'ing all the time, literally none of which you need to actually worry about under normal circumstances, as you can generally just hold down the trigger and everything will die, but it does also mean that if something goes wrong in your rotation then figuring out how to get it back on track can be extremely annoying.

The Basics

Role: Main DPS
Deals Thermal Damage
Weapon Type: Dual SMG
Deals entirely Ballistic damage
A lot of her DMG counts as skill damage though...
Mobility is dia-fucking-bolical

How to use Divine Fox

Against Single-targets / bosses

If her Ultimate Skill is available, use it.
Otherwise, just hold down the trigger.

With her Standard Skills Neuronic buff, she regains ammo naturally, so you don't normally need to actually reload - just hold down the trigger.

Against groups

1. If her Ultimate Skill is available, use it.
2. Spam her Standard Skill off-cooldown
3. Otherwise, spam her ADS button to deal AoE damage.

How to build Divine Fox

Weapon:
Porcelain Grace / Petal Dust
Weapon Parts: Max ATK
Logistics Squad: Moon
Basic: +10.0% ATK
+106 Alignment Index
Special: +8.5% Thermal Damage
Weapon Parts: ATK
Good Support Combos to support Longyan with:
Tess - Phantasia + Yao - Nightglow
Tess - Phantasia + Marian - Aquila
Tess - Phantasia + Fenny - Starshine
Tess - Phantasia + Fritia - Little Sunshine
Vidya - Celestial Swan + Fenny - Starshine
Vidya - Celestial Swan + Fritia - Little Sunshine

Standard Shot / Passive Effect

Whenever Longyan fires or uses ADS, she enters the "Brocade Pelt" status.

Whilst in Brocade Pelt status:

Left clicking causes Longyan to shoot at the target.
Right clicking causes Longyan to deal AoE damage, hitting up to 5 targets.
This is a shot, but doesn't use any bullets.
Longyan gains 1 "Fox Favor" per LMB shot fired, and 10 per every RMB shot fired, up to a max of 60.

For both of these, they increases the damage of Elemental Aptitudes to 300% of their normal damage.

Longyan literally cannot go ADS - attempting to go ADS will instead trigger the corresponding ADS proc.

After shooting 3 times in a row, the next usage will perform an extra attack, which:

Counts as skill damage, but also counts as a shot, and hence does proc Aptitude effects.
Deals more damage (of the same type as the attack that triggered it, so LMB deals extra single-target, and RMB deals extra AoE)
Always crits
Doesn't use any bullets.
Increases the damage of Elemental Aptitudes to 1,000% of their normal damage.
With its first Neuronic, fully replenishes her magazine.
Consumes all Fox Favour, decreasing her Standard Skill Cooldown by 0.05s per Fox Favor.
With her second Neuronic, it also gains 1% Final DMG per Fox Favor, which is potentially a very strong buff.

Longyan's Deiwos will give enemies a "Cinder Mark" after Longyan damages them whilst in Brocade Pelt status.
When enemies have 30 Cinder Marks, they are removed to deal some Thermal DMG and give Longyan 1 Tail Flame (which is used for her ult).

Standard Skill

Longyan dashses forward and deals Thermal Ballistic DMG to up to 5 enemies.

This is skill damage, but also counts as a shot
This is a guaranteed crit
This doesn't use any bullets.
Increases the damage of Elemental Aptitudes to 1,000% of their normal damage.

Longyan also gains 2 "Tail Flames" which are used for her ultimate.

Ultimate Skill

Longyan's Ultimate uses 0 U-Energy, but instead she needs to have 9 Tail Flames (from her Standard Skill) to use it.
Note that each use of her Standard Skill gives her 2 Tail Flames, but she also gets lots of them from her Deiwos.

Longyan uses all her Tail Flames to deal damage 8 times, followed by 1 stronger shot, which are all Ballistic Damage, and can damage up to 10 enemies.
These deal Ballistic damage, but are a skill usage, but also count as a shot.
Like much of the rest of Longyan's kit, these always crit, don't use bullets, and increase the damage of all her Aptitudes [to 1,000%].

Much like a lot of the recent operatives who have Ultimates that deal AoE damage, with its second Neuronic effect, Longyan's ult deals more damage the fewer targets they are, dealing +15% Final DMG per target less than 10, which is a huge damage boost against bosses.

Support Skill

Longyan deals AoE damage in front of her.
If it hits a target, it will proc another, large AoE, which can itself proc another even larger AoE if it hits.
With its Neuronics effects, the first and second stike will decrease movement speed for 2s, and the third strike will also Knock Back the targets.

Like all Main DPS operatives, Longyan's Support Skill isn't very good, and you really don't want to be using her as a support operative.

Weapon

Porcelain Grace
Her signature 5* weapon, and her best weapon.
This gives ATK and Thermal DMG, but as usual for a 5* weapon, the most useful part of the weapon is the "Thermal DMG ignores X% of target's Thermal Resistance".
Petal Dust
Increases her ATK and Thermal DMG.

There is no other operative who uses Thermal-damage Dual SMGs, so if you don't have either or Longyan's signature weapons, then you can use one of the signature 4* weapons of another Dual SMG operative if you wish, but since none of them will proc her Deiwos effect or several of her Manifestation effects, they're all much worse than either of Longyan's actual signature weapons.
The default 3* Dual SMG that Longyan comes with (Optimal Solution) doesn't perform that well for her, as it isn't a Thermal-damage weapon, and hence doesn't proc her Deiwos effect or seeral of her Manifestation effects.
Bubu - Specular Soul's 4* weapon (Serrated Pines) does out-perform the default 3* weapon Longyan comes with. Bubu's 5* doesn't work well with Longyan.
Longyan doesn't work well with either of Lyfe - Infinite Sight's signature SMGs.

For weapon Parts Longyan wants max ATK.

Logistics

Squad

Use Moon squad - it was designed specifically for her and is her best choice.

Moon squad gives her 20% DEFpen when she has an Elemental Aptitude.

When she's in the Brocade Pelt state, she gets +40% Thermal DMG
When not in the Brocade Pelt state, she gets +40% Standard Skill and Ultimate DMG.
Since Longyan is basically always in Brocade Pelt status whilst she's on the field, this is pretty much just a constant +40% Thermal DMG buff.

Stats

Longyan wants:

Basic Stats +10.0% ATK
+106 Alignment Index
Special Stats +8.5% Thermal DMG

ATK to maximize her damage.
Alignment Index gives her more ATK - a max-roll of +106 Alignment Index translates to gives Longyan +5.3% ATK.
Thermal DMG for more damage.

Longyan can't heal, create shields, basically all of her damage always crits anyway, she doesn't use U-Energy, and doesn't normally have any S-Energy issues, so most stats aren't useful to her.

Crit DMG Amplifier might seem like a good stat since Longyan always crits, but her Dual SMGs have a base Crit DMG value of just 30%, meaning that Crit DMG Amplifier mods don't actually boost her damage by that much.
With three maxed Crit DMG Amplifier mods, Longyan's crits would go from 130% to 139.54%. This is not a massive increase to her damage, and she's generally better off getting Alignment Index to get the +15.9% ATK that three max-roll Alignment Index mods will give instead of getting ATK.
Crit DMG Amplifier is a good stat for Longyan, but Alignment Index is just a bit better.

Manifestation

M1 gives Longyan's ballistic DMG ignores 15% of the enemies Thermal Resistance
This only works if Longyan has a Thermal-damage weapon equipped, but that should always be the case.
This is a pretty decent damage buff.
With M2, Longyan damaging an enemy with Thermal DMG reduces their DEF by 5%, up to 20%.
This is a good damage buff.
With M3, Longyan's Ult increases its Final DMG by 8% per hit, which resets after the ult finishes.
This isn't quite linear, as her 9th/final shot deals more damage than the first 8.
Overall, M3 works out to a 43.8% Final DMG increase for her ult, which is strong, but it's only for her ult.
M4 gives the usual +10% Damage to her Ultimate
This is a decent damage increase, but M4 is pretty expensive, and it's just a damage increase.
M5 increases Longyan's ATK by 10%, and increases the ATK of your entire squad by another 10% if Longyan has an Aptitude effect.

If you don't want to get her all the way to M5, then I'd say M3 is a good stopping point.

Good Supports for Longyan - Divine Fox

Tess - Phantasia works perfectly with Longyan.
Tess provides some great ATK and Ballistic damage buffs, which affect all of Longyan's damage.
Longyan deals damage very frequently, which gives Tess' Aptitude effect a lot of chances to add extra damage.
Longyan increases the damage Tess deals as well.
Tess and Longyan work perfectly.
Vidya - Celestial Swan
Vidya massive reduces the enemies DEF and Resistance, and buffs the teams damage a little too.
Vidya only debuffs the damage resistance of the damage type the enemy has most recently received.
As such, if using Celestial Swan, then make sure the other non-Vidya support isn't off-element to Longyan.
(i.e. They either deal Thermal damage or do not deal damage at all).

Whilst Vidya and Tess are Longyan's best supports, since Longyan deals Thermal damage and Tess deals Frost damage, you can't support Longyan with both Tess and Vidya at the same time.

Other great supports for Longyan include:

Yao - Nightglow
Yao buffs ballistic damage, ATK, and gives extra Crit DMG Amplifier, which are three stats Longyan really wants, so Yao adds a lot of damage to Longyan.
Longyan has quite a low base Crit Damage value of just 30%, but even with that Yao is still very strong.
Yao is off-element to Longyan (Yao constantly deals Electric damage).
Marian - Aquila
Aquila provides very strong generic damage buffs, which makes her an excellent support.
Aquila is off-element to Longyan though (Aquila constantly deals Kinetic damage).
Eatchel - The Cub
A Blitzing Fangs Eatchel provides healing as well as a decent ATK buff.
Not the strongest support, but still good, and Longyan's horrendous mobility means she really appreciates Eatchel's constant healing.
Fritia - Little Sunshine
Whilst she's only a 4* operative, Fritia can do a lot of hits for Longyan, due to Longyan's extremely high average Rate-of-Fire.
Longyan also inherently boosts the damage of Aptitude effects, including Fritia's.
Fritia is also on-element to Longyan, and can thus be used alongside Celestial Swan.
Fenny - Starshine
Starshine provides a lot of very strong ballistic damage buffs, however Fenny's buffs are only active when the deployed operative has one of her Special Clips equipped.
Having a special Clip equipped though prevents Aptitude effects from working, so Starshine's buffs block Tess - Phantasia's Aptiutude damage, which is a huge loss.
Starshine is thus mutually exclusive with Phantasia, however Phantasia is so strong that this means that you'll only want to use Starshine when not using Phantasia.
Vidya - Celestial Swan + Fenny - Starshine is a great combo, but it is out-performed by some of Tess' combos.
Acacia - Kaguya
Kaguya is a good resistance debuffer - not nearly as strong as Vidya - Celestial Swan, but if you aren't using Vidya then Kaguya can be used as a much weaker substitute.
Kaguya deals Frost-damage.

Longyan benefits from all of the buffs that Tess - Phantasia can provide, whilst also boosting Tess' Aptitude damage as well, and Phantasia is thus Longyan's best support, and since Tess and Vidya are mutualls exclusive, Longyan's best support pairs are thus:

Tess - Phantasia + Yao - Nightglow
Tess - Phantasia + Marian - Aquila
Tess - Phantasia + Fenny - Starshine
Tess - Phantasia + Fritia - Little Sunshine
Vidya - Celestial Swan + Fenny - Starshine
Vidya - Celestial Swan + Fritia - Little Sunshine

Cope Supports like Mauxir - Shadow Ka, Vidya -Agave, Acacia - [Redacted], Yao - Quiet Quitter, etc. are still fine if they're all you have.