1) If I do it quick and dirty, no corner bracing, using 4' fence on 6' T-posts sunk 2 feet, will it flop over? I know that 4' is not tall enough to keep elk out, but I was thinking of then adding bamboo poles to the corners and running fishing line or trellis netting above the 4' fencing. Of course this will depend a bit on the weight of your 'fencing' materials, the heavier the material the more braces/posts you'll need, in the corners and along the line. I will assume you are going to use wielded-wire or something lighter, and that you will NOT be wrapping your wire around your corner posts - in this case T-posts every 6' to 8' are fine, with one used for the corners. If you are using only chicken wire you can get away with plastic, metal stakes instead of T-posts.
T-posts that will have extra things pulling or leaning on them will move, so those used are doorways or corners with rope/twine will need some type of bracing. This doesn't have to be extreme just what's needed for the particular situation.
2) If I -do- need bracing on the corners, what is the best way to go in getting it done alone and rapidly? I have seen the clips that allow extra T-posts to be diagonal and horizontal braces, but they are quite spendy and hard to find in this area. Wooden posts seem like overkill for such a small area. First, proper fencing depends much more on the critter and use than on the area size to be fenced.
I agree with the over-kill feeling. Deer don't throw their weight around like say a goat, bear or even a dog running at a target - this is good news for your situation, as you say you can add bamboo polls (I used recycled floor molding strips for the same thing) and run a light weight netting or rope to address the jumping issue. Since you'll be pulling this tight, some bit of bracing will prevent your corner posts from leaning.
If you don't have kids, animals, etc. that will lean on your fencing then you could use lighter/cheaper plastic/metal stakes instead of T-posts, and only use the heaver T-posts in the corner bracing..... Or find a source of recycled materiel for free that will meet your bracing needs while saving you some of the $.
A nice looking fence, that needs to stand up to many different things, such as neighbors pets/animals and children as well as yours, should have corner bracing. If you are really only addressing deer and your trees then I would only protect those - saving time and money.
3) Would it be better in this case to just make rings around each tree? IMO - Yes