Is it possible to increase/train your intelligence? Is it possible to raise your IQ (best correlate for g I know of)? Does brain training help? 

I have never taken a psychologist administered IQ test before, but I did take a reputable online IQ test and experienced a shift of 20+ IQ points over a two/three year period.
I didn't do the tests incessantly, did it once, then revisited the test a few years later, and I was testing 20+ points higher than my previous score.
Here is the first article that popped up on a Google search of your first question. It poses your questions to 5 "experts" (psychology professors).

The subtlety in their responses comes from the fact that an IQ measurement is an imperfect number meant to capture some sort of underlying construct (and that this underlying construct can constantly be under debate). So, when one raises one's score: did you get smarter, or just become a better test taker? I won't answer one way or the other, but I will quote one of the experts (discussing IQ training with children):


  ... what I would say, is we didn't make the children smarter, but we taught them how to use what they have more efficiently, and better.


So yes, you can raise your IQ score. Now what does that imply? That depends on whom you ask...