Can I use Fusiler Dragon TD-MB's effect to summon him if Skill Drain is on the field, and if I can, would his atk/def still be halved, because if I'm correct, his attack and def can't be halved until he's on the Field, and when he's face-up on the field, skill drain would negate his effect. Is that correct?

Also, if I use bazoo The Soul-Eater's effect, and then active Skill Drain, would he get to keep his atk increase?

Other posts already explain Fusiler Dragon and Skill Drain. Yes, he will not be halved with Skill Drain on the field.

No, using Skill Drain would negate Bazoo's overall effect so you would end up removing 3 monster cards for nothing since Skill Drain will make its attack 1600 again.