Katya - Blue Bolt

If you don't have Katya - Blue Bolt yet but want to try her out, she is available as a Trial Operative in Story Chapter 12A-7.
She is also available in most of the missions in chapter 12B "Skyward Blaze" (Legacy // Yehrus // Skyward Blaze).

Blue Bolt can shoot at a high rate, never needs to reload, and can quickly and easily swap between single-target and AoE shots, but she needs to set up to deal high dps which gives her literally zero mobility, and in the time since her initial release her damage has been powercrept by newer operatives.

Overall

Role: Main DPS
Frost Damage
Weapon Type: Crossbow
Deals entirely ballistic damage
Note that Blue Bolt can't crit.
Mobility is zero for most of combat

How to use Katya - Blue Bolt

Reload to swap between shots that deal high single-target damage and shots that deal AoE damage.

If Katya has S-Energy: Use her standard skill and hold down the trigger to shoot enemies until she runs out of S-Energy
If Katya doesn't have S-Energy: Just shoot enemies.
With Manifestation #3, also dodge every 5s.

Repeat

How to build Katya - Blue Bolt

Weapon: Nepture Nova / Alpine Gentian
Weapon Parts: Max ATK
Logistics Squad: Eli
+10.0% ATK
+106 Alignment Index
+8.5% Frost Damage
Best teammates:
Vidya - Celestial Swan
Tess - Phantasia
Yao - Nightglow

Passive Effects

Katya uses a crossbow.

Drop-off and Terminology

The bolts from her crossbow to have significant drop-off over distance, and are very much not hitscan weapons like the guns of other operatives.
Note that linguistically an "arrow" is the projectile shot (not fired) from a bow, whereas crossbows shoot "bolts", however SCZ usually refers to Katya's bolts as just "shots".
(Guns fire their bullets because they burn gunpowder to do it. Bows and crossbows do not use gunpowder, and are thus not fired - they are shot.)
(Also; yes, modern [smokeless] gunpowder is not an explosive as it burns at below subsonic speed. Gunpowder is thus a deflagrant, not an explosive.)

Crits and Crit Damage

Crossbows can't crit, even if shooting at a weakspot.
Even if they could crit, all crossbows have a 0% Crit Dmg value.
This means that Katya's crits would always deal the same damage as her non-crits anyway.

In SCZ, the Crit DMG Amplifier stat only add to the Crit DMG stat of the operative.
For example if an operative has 60% Crit Dmg and +50% Crit DMG Amplfiier, then their crits wil deal 190% damage (100% base, plus 60%*1.5 for the extra from the crit)
Since Blue Bolt has 0% crit dmg, Crit DMG Amplifier mods wouldn't do anything for her, as anything x0 = 0

Since Katya can't ever deal actually deal extra damage via crits, the support buffs from other operatives that give crit buffs will instead give ballistic damage to Crossbow users.
For buffs from other operatives, how much Crossbow users get will always be explicitely stated.
For unspecified buffs, Katya gains 1% ballistic damage per 1% crit rate, and 0.6% ballistic damage per 1% crit damage amplifier.

Infinite Ammo and Ammo Types

Katya's crossbow has infinite ammo and never needs to reload.
If you press the reload key, Katya will instead swap between "Normal firing mode" and "Special firing mode".
Normal firing mode fires single-target bolts. By-default these fire at a rate of 150 shots/minute (2.5 shots/second).
Special firing mode fires bolts that have a lower RoF but deal AoE damage (to up to 5 targets). By-default these fire at a rate of 85 shots/minute (1.41 shots/second).

Passive Extra Damage

Katya also passively gains a 35% chance to gain extra 35% Frost damage for each of her shots.
Plus an extra 7% damage per 100 Alignment Index.
Whilst shooting AoE bolts, the chance is doubled, but the damage bonus is halved.

This extra damage deals double damage against Frozen enemies.

This extra damage counts as skill damage.

Standard Skill

Use Katya's Standard Skill to toggle "Guidance III" form, which locks her in place - she becomes completely immobile, but can still turn freely.
This massively increases her Rate-of-Fire (to 8/s for Normal mode, and 2.86/s for Special), but decreases her damage, destructible object damage, and give less U-Energy.
Normal shots still do less damage than Special shots.
Her higher RoF is fixed, and is thus unaffected by RoF buffs.
Katya also gains +15% damage resistance and CC-resistance.
Dodging leaves this form.

Whilst in this mode, Katya can't recover S-Energy, and each shot uses S-Energy (0.5 for Normal firing mode, 1.5 for Special).
...but when not in this mode, her Neuronic buff gives her 2 S-Energy per hit, allowing her to quickly recover it.
(This is unaffected by S-Energy Regen mods.)
When S-Energy runs out, Katya automatically leaves Guidance form.

Entering and leaving Guidance form doesn't use any S-Energy, and it only has a 3 second cooldown.
The cooldown starts when she leaves the mode.

Ultimate Skill

Katya deals AoE damage in front of her, dealing damage every 0.4s for 10s.

With its second Neuronic buff, each hit reduces the targets movement speed, and inflicts Frozen to enemies that are hit 10 times.
This helps proc her Deiwos passive, which doubles the damage of her passives aptitude buff against Frozen targets.

Support Skill

Katya fires a projectile that deals AoE damage and inflicts Frozen for 3s.
This knocks Katya herself back a little.

Like all main dps operatives, Katya's support skill isn't very good, and you don't want to use Katya as a support operative.

Weapon

There are only four crossbows in the game:

Neptune Nova
Her 5* weapon, and her best weapon.
When stationary, recovers 0.1/0.15 S-Energy per bolt that hits.
Each bolt normally costs 0.5/1.5 S-Energy, so this gives another 25%/7% bolts for her standard skill for T1, and 42%/11.1% for T2, which is a good sustain buff.
Alpine Gentian
Burbank Sky
Resistance

The higher rarity ones are obviously better than the lower rarity ones.

For weapon Parts Katya wants max ATK.

Logistics

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

Eli gives a stacking Frost dmg buff after a non-crit, with crits removing stacks.
Since Katya can't crit, this is great for her as she can't lose stacks except by them timing out.

For stats Katya wants +10.0% ATK and +106 Alignment Index.
For the Special talent, Katya wants +8.5% Frost Damage

ATK is Katya's best stat by a huge margin.

Alignment gives more per-shot aptitude damage.
Three maxed-roll Alignment Index mods will give an extra 22.26%.
This is better than any other non-ATK mod.

Crossbows gain 1% ballistic damage per 1% crit rate, so a max-roll of 5.3% Crit Rate will give Katya +5.3% ballistic damage, which is a good buff, but not ideal.
10.6% Crit DMG Amplifier will give Katya 6.36% ballistic damage. Also good, but also not ideal.

Whilst it might seem that Alignment isn't much better than crit rate or damage, Alignment does work out much better for Katya.
To simplify a bit: Katya will gain a lot of damage mods from other sources, so it's better for her to have the extra base damage from Alignment for all the other mods to multiply off of, rather than for her to have the extra ballistic damage from the crit mods, which would give her a much lower base damage with a slightly higher damage modifier, which does end up worse.
Crit mods are still good stats for Katya - they're just not as good as Alignment Index.

Since Katya can't recover S-Energy during her standard skill, S-Energy regen isn't overly useful for her.
It can help to recover S-Energy after she runs out, but you can just shoot enemies to do that, which you'll be doing a lot of anyway.
Katya shoots at a rate of 2.5 shots per second when not in her standard skill, and each shot gives 2 S-Energy, so she gains 5 extra S-Energy per second anyway.
(for 6 total, as she has a default of 1 S-Energy recovery per second like all other operatives.)
Having three S-Energy mods will thus take her from 6/s to 6.423/s - it's not a massive increase, and they aren't nearly as useful as just giving her more damage.
S-Energy regen can help, but is a weak cope stat.
U-Energy Regen is technically useful, but since Katya's ult isn't that good and she doesn't recover much U-Energy, U-Energy Regen isn't a good stat for her.

Eli is Katya's best logistics squad, but Navigator is also very good due to how often Katya can Freeze enemies.
Mizuho is also great as Katya doesn't do much other than stand around and continuously shoot things.

Manifestation

M1 gives Katya +1% RoF per shot whilst using her standard skill, up to +25%.
This puts her up to 10 bolts/s and 3.575 bolts/s for Normal/Special mode, respectively.
Katya loses 5 per sec whilst not shooting, and loses half the stacks when damaged up to every 0.5s.
This is potentially a really nice damage buff (of +25%) but taking damage will massively reduce its effect.
M2 gives Katya's standard skill +30% destructible object dmg and +0.8 S-Energy per bolt hitting a destructible object.
This is a great S-Energy regen buff if shooting at a destructible object whilst using her standard skill.
If not shooting a destructible object, M2 is completely useless.
M3 reduces the stamina cost of Katya's dodge, and prevents the stacks from her M1 from decreasing for 5s after dodging.
This means that you can just dodge immediately after her standard skill ends, and then keep her RoF buffs for much longer.
You can dodge every 5s to maintain the buffs indefinitely if you want.
M4 increases the level of her standard skill, increasing its damage a smidge.
This is just a damage increase.
M5 gives Katya +165 Alignment Index.
Since 100 Alignment Index gives her +7% to her aptitude buff, +165 Alignment Index gives her +11.55%.
This is just more damage per shot.
Given that Katya's crossbows have high Compatibility and naturally has her own aptitude that deals extra damage on top of that, M5 isn't a massive damage increase.
It's good, but it isn't a massive percent damage bonus.

M1 is a very nice dps increase.
M2 and M3 are more situational and might not be overly useful in many situations.
As such, if you want efficiency (good damage without massive investment), M1 is potentially a good stopping point, but M2 or M3 might also be good depending on what type of content you are using Katya for and what enemies you tend to face.

Good Supports for Katya - Blue Bolt

Katya deals entirely ballistic damage at a high RoF.
This means that she wants ATK buffs, ballistic damage, Frost damage, per-shot aptitude buffs, and that's about it.

Vidya - Celestial Swan (and other resistance shredders)
Celestial Swan and other resistance shredders (Acacia - Kaguya, etc.) that reduce the enemies resistance help enemies take more damage.
If using Celestial Swan, make sure the other support isn't off element to Katya.
Kaguya is also on-element to Katya, allowing Katya to benefit from Kaguya's Frost buffs.
You could also use Kaguya with Celestial Swan, although that's not ideal.
Tess - Phantasia
Katya fires at a very high rate, so the per-shot damage from Tess works out very well for her.
Phantasia is also on-element to Katya, allowing her to be used alongside Celestial Swan.
Fritia - Little Sunshine works as a weaker off-element cope version, but does still work quite well due to Katya's very high fire rate.
Yao - Nightglow
Yao provides a strong ATK buff and several ballistic buffs, making Yao extremely strong for Katya.
Note that Katya gains extra ballistic damage instead of the crit mods that Nightglow usually give.
Nightglow doesn't work alongside Celestial Swan, but Nightglow is otherwise extremely strong for Katya.
Eatchel - The Cub
Katya's horrendous mobility means that Eatchel's healing can be very desirable. Eatchel's ATK buffs are also nice.

Katya's best supports are Vidya - Celestial Swan and Tess - Phantasia.
Kaguya + Little Sunshine/Nightglow does also work well should you not have Vidya and Tess.
Eatchel works with all the other supports you'd want to use with Katya.