Assault Mode Activate vs Skill Drain

I chain Assault Mode Activate when my opponent activates Skill Drain to bring out Doomkaiser Dragon Assault Mode. Does Doomkaiser Dragon Assault Mode get its effect before Skill Drain resolves?

No it doesn't. Doomkaiser's effect will resolve after the chain, but by then skill would of already successfully hit the field, so no effect.

So you have your current chain which ends with assault mode activate. Now what happens is that chain resolves backwards. Assault mode resolves, tributes the synchro and summons the new monster. Nothing can happen here until Skill drain finishes resolving. Now that the current chain is finished, Doomkaiser's trigger effect will activate here on it's own link. It will still be negated by skill drain however.