When answering questions on this site, are there any particular criteria you personally employ to determine whether you answer a certain question or not?

Not really. If I can answer it, I will.

I love finding questions that send me on a unique research. 

I'm in the minority, but I like homework questions.  Guiding students to learn how to search online, or solidify their knowledge is very satisfying.  I won't tolerate lazy students who just want to cheat.  (Sometimes they just lack confidence.)  But helping and tutoring is a delight, especially when providing a visualization or unique explanation that brings understanding.  The best gift of teaching was seeing the expression on their faces when they truly got it.  I can't describe the feeling that washes over a teacher when their eyes go wide and their mouths slightly open.  They get it! 

I also enjoy the fun ones that engage the community in lots of conversation.  It's actually trickier for me to come up with those conversation starters.  I think asking is harder than answering. 

I guess I'm just glad to have a Q&A community with so much variety, and such neat members. 

I always look for "value added". If I can include a quirky fact, some humour, or a lateral approach to the question, I try to do that.

If the answer really only comes from Google/Wikipedia (i.e, I could not add anything that the asker couldn't find out easily themself) I usually leave those alone.

I'm less likely to answer a question if it has been written by Mr Anonymous.

I never answer medical questions, I'm not a doctor, and despite the website disclaimers, there is always the risk that I give bad advice, someone takes it and they suffer.

Like Happy, I don't mind "homework" questions, provided that the asker is looking for help and not just copying the question verbatim.

And, if you look at my profile, you'll see I'm into astrophysics and English dark-ages history. Sadly, we get very few questions on either of  those (although historically we used to get a lot of "astrophysics" questions).

Wasn't that boringly pompous and serious? Back to normal now.