What's the best approach to capturing Lumine? Is it possible to do before the triple stat increase occurs?
It is very possible to capture her on week 40. It might be possible to go earlier, but I don't think you can get a Cryo Samachurl before week 20. A Dendro Samachurl, Cryo Samachurl, and 6 Nycto Hilichurls make quick work of her. Dendro mines the field, preventing her from aproaching your forces and building up her elemental guage. Cryo pillars the Dendro and Nycto, improving their damage and extending their range. Nycto bring the pain.
Key characters to capture for this strategy;
Patchi - Adventurers for Shielded Hilichurl - Civilian at Thousand Winds Temple
Anna - Adventurers for Archer Hilichurl - Mondstadt Boss at 20 Threat
Dr. Livingstone - Scholars for Dendro Samachurl - Civilian at Windrise
Esther - Scholars and Cryo Hilichurl for Cryo Samachurl - Civilian at Dragonspine Camp
Amber - Pyro Hilichurl production - Mondstadt boss at 50 threat
Rosaria - Cyro Hilichurl production - Mondstadt boss at 150 threat
Collei - Dendro Hilichurl production - Starts approaching after Amber defeat.
Barbara - Nycto Hilichurl production - Mondstadt 1
Sucrose - Offspring potion - wanderer hidden at Snow Covered Path
Scholars - Windrise, Thousand Winds Temple, Dragonspine Camp
Falcon Coast - Fischl's hiding place - NOTE: Fischl could be a good target for corruption, but she gets better if you let her escape. So it might be better to ignore her for now, since we won't have a lot of bandwidth for keeping captures.
Basic outline -
The fastest way to get a cryo samachurl is vai Esther's training. That training needs basic cryo hilichurls. To get 20 of those for the initial training, within the 40 week timelimit, we'll need both Anna's training to pull Cryo's, and a Cryo heroine to spawn them. That'd be Rosaria. We could also breed non-cryo heroine's, but you get better results with a Cryo. Since Rosaria is rather powerful, we'll need powerful units to fight her - meaning Dendro/Geo Hilichurls. We can get those from training with Patchi, or via breeding Collei.
Some General fight tips:
Week 1 Lumine; Good practice for positioning. She has no ranged attacks, so if you move just outside of her movement+attack range, she'll only be able to move into range. A round of Attacks+minion captures with all 4 of your units should be enough to capture her.
Amber; Likes to drop her bomb 2 hexes away from her, and between the highest number of units. You can use this to direct where she'll move. Basically, group up your units so there's one good spot to drop the bomb, and she'll move into position for you. Alternatively, I've also seen her just move and shoot on her first turn. If she does that, then you can easily caputure her afterwards without taking any real damage. Otherwise, use the hilichurl charge attack to knock her away from the bomb, then swarm her. If you bring 6+ units, you should be able to capture her on round 2.
Collei; her AI seems to be a bit temperamental, so her behavior is a little inconsistent. She likes to use here AOE attack to hit as many targets as possible, and has a preference to attack the furthest units possible. So if you setup you forces in a triangle formation, one hex between each unit, she'll move next to the center of your formation to attack the center + the backline. She has a really OP area effect burst that will cause a lot of damage if she gets it of, and you can potentially get her to retreat if she takes enough damage. Basic strategy is to use two rows of units in a triangle formation to bait her forward, then charge+swarm to drop her HP enough that she triies to retreat. Failing that, use charge attacks to force her out of the field effect, and keep hitting her till she's down. Try to face her with at least 6 basic Hilichurl. 8 would be better, but she's a tad reluctant to actually attack a force of 8, and you need to bait her into attacking your position at Dragonspine. Alternatively, you might be able to bait her by retreating from Dragonspine, then attacking it when she moves in.
Basic moves:
Week 1 - build another Hilichurl squad, start wedge production
Week 2 - Capture Windrise, Reinforce/replace till you have a squad of 6. Keep up Hilichurl production.
(Capture Anna)
Week 3 - destroy statue to spike threat to spawn Amber. Capture Thousand Winds Temple, retreat main force back to Windrise. Maintain squad of 6, potentialy go up to 8.
(Capture Amber) (Collei starts coming over)
Following weeks - build up squad of 4 to take Dragonspine camp, station 6 there to take Collei when she attacks. Capture as many adventurers as possible, use them to train Cryo and Dendro/Geo Hilichurl with Patchi and Anna training.
Collei should arrive to attack around Week 8-10, letting you start real Dendro hilichurl production.
Once you've got ~4 shielded hilichurl stationed in Windrise, you can attack Mondstadt to get Barbara, and destroy the two trees you can reach to spike threat high enough to spawn Rosaria. You should be able to do this before week 20. Barbara doesn't have enough hydro power to be worthwile, so she's a good target for corruption. Once corrupted, she'll be able to start producing nycto units for you.
Finally, if you need any more units, capturing Rosaria + getting the Crimson Agate should give you enough Cryo to enter the Snow Covered Path. Doing that should spawn Sucrose, and capturing her will let you make her potions. Which will boost the amount of offspring your heroines produce.
Notes on the farm and alter;
Wedges are the most cost effective way of generating abyss power early on. I'd suggest planting 5 of them early on. That won't generate enough threat to be a problem, but will generate a steady trickle of abyss power for you.
Also, you'll want 3-5 slime farms early on. sacrifing 10 nycto slimes generates 1 abyss power. If you've planted the abyss wedges, slime farms will generate nycto slimes. It's not much of an income, but it is a way to generate consistant abyss power without generating threat.