Favorite deity is probably Azothoth, just because his/its description makes me quasi-suspect it's really the ET version of an anthropomorphized natural force or principle rather than a literal entity.
Most useful is Nyarly, 'cause he's a bit of a swiss army knife, what with his unlimited forms, inscrutable ways, and position as sort of the Apollo of the mythos pantheon. That same flexibility actually makes him boring to me though: his lack of definable anything makes him shallow as a character, and too transparently just a universal plot device. A diobolus ex machina, if you will.
Favorite critter is the Mi-Go.
Whisperer in Darkness is probably my favorite Lovecraft story, because I love how despite everything you pretty much end up know nothing for certain at the end: anything or everything the protagonist has seen and heard could've been a lie. The mythos stuff written by later authors, especially the games fluff, tends to assume it's almost all true, but in the story any combination of lies and truth is equally possible, up to and including the possibility that the whole thing is a hoax and the Mi-Go don't exist at all.

That would seem to make them just as much a Swiss Army Plot Device as Nyarlathotep, but it doesn't: the Mi-Go have got hints of definable form and personality for the imagination to latch onto and run away with. Personally, I like to picture them as affable tricksters (or rather they're not, but that's how they come off to humans because of psychological differences), but anyone elses interpretation is just as "true".
I also like the Polyps for similar reasons: the only things we "know" about them are based on Yithian propaganda, and that doesn't 100% add up with what the protagonist sees/experiences in the latter part of the story. But on the other tentacle, The Yithians seemed to have a somewhat laissez-faire attitude about sharing info, so it may not have been in character for them to lie as such...