Strange is trying to make himself mad so that he can see a fairy - I can't actually remember if and where they explain that but I assume they did! There was some chat in the episode with King George about fairies having a connection with mad people. Basically, he thinks (rightly) that being mad will help him summon a fairy.
Paul Kaye - can't explain anything without giving away spoilers, that part of the story isn't finished yet
The Gentleman just thinks that Stephen is no noble and handsome and wonderful that he ought to be a king. The Gentleman is not a rational man 
Segundus and Honeyfoot are running a madhouse, so it's not "their house". (When he stopped them from opening a school in that building, Childermass said he'd help them in any way he could in the future, and so when a madhouse was needed he recommended Segendus)
Drawlight - I don't think they could have sent anyone, they needed someone who understood Strange, but also Drawlight is very good at ingratiating himself and finding out gossip, as in he has no standards and will just demand people tell him things.