I've been reading a lot of Urban Fantasy lately, and most of them are in first person, and I've been really liking it. I think a lot of it depends on what kind of story. Is it a smaller story about 1 person specifically, or is it a big story that really needs a wide scope to tell it? If it's the former then I think it first person is OK, but for the former I think third person works better. Which probably why you don't find a lot of first person stories in Trek, they tend to be bigger in scope than just one person's experiences.
That's very much the case. First-person is probably better suited to smaller, more intimate, possibly more introspective stories than, say, an epic about the Fall of Rome. In the case of Star Trek, where you have an ensemble cast and are usually cutting back and forth between various A, B, and C plots, third-person is usually the most sensible choice.
It's all about using the right tool for each job--and what you can do with that tool that you can't more easily do with another.
In the case of the "In the Grove," the whole point of that story was to illustrate how different people view the same event differently, so first-person was ideal there.
I find that a lot of YA books are written in first-person these days, with the Harry Potter books being a notable exception. Not entirely sure why that is unless adolescents prefer a tight focus on one character and his/her feelings. (See Twilight, which I believe is written in the first-person.)