A) Who is your favourite Pokemon Professor? Why?
Does Professor Ivy count? 8D /kidding
My vote would have to go to Professor Oak. This is mostly manga-influenced, because if you just see what sort of things he does in the manga, you would see how freaking awesome he is. Though if I weren't to count him, I'd probably say Professor Juniper, since she seems to do a lot more than the other Professors (though Rowan and Oak also do plenty, especially the latter since he actually travels to other regions).
B) What would be one thing you would research if you were a pokemon professor?
Pokemon natures and traits. I've got natures down pretty well (traits, not so much), and both are pretty vital in the world of competitive battling (although not so much traits because they give a hint as to what stat has the most IVs or something, right?). And, to a lesser extent, regular battling, I suppose. Though it isn't really that hard to beat the E4 with an Adamant Gardevoir unless it has a super low Sp. Atk. IV or something. But then there's the Battle Frontier/Subway, which sort of relies on certain competitive battling aspects. So yeah.