Applied researcher with deflect and repeat costs

Yes — while it's Empowered, Applied Researchers' discount offsets Deflect.

Costs are worked out in order: base cost, then Deflect's mandatory additional Power cost is added (Rule 356.2, 809.1.c), then discounts are applied to the total (Rule 356.4). Since discounts can reduce additional costs to 0 (Rule 356.4.f), the "[1][A] less" from Applied Researchers covers a Deflect 1 surcharge — you pay your normal cost, not the extra Power, when your spell chooses a Deflect unit.

Two limits: it only applies to spells you play (not your activated abilities, and not reflexive triggers your spells create), and it only works while Applied Researchers is Empowered (pay [3] first). It also doesn't bypass Deflect's targeting rule — the cost is still added and then discounted, and if Deflect is higher than 1 you still pay the remainder.

I noticed multiple unrelated questions. I've answered the first one — please ask the others in separate messages so I can give each one proper attention.