I never camp under or near a tree for all of the reasons stated, I always try to pitch near a hedge to protect from the wind and I also don't like pitching in the middle of a field.
We always try to avoid pitching under trees due to all the aforementioned reasons, plus if the wind gets up, it whistles through the trees & sounds much worse than when you're camped in the open, and it certainly keeps you awake at night with the whistling & the russle of the leaves too. Yet it's strange but I love to pitch near a stream, and I find that the sound of the water running over the stones helps me to sleep
We camped on a site where unbeknown to us, a family of crows were roosting in the tree above.
A rude awakening in the way of constant noise at silly o'clock in the morning and bird poo falling on the tent has put me off camping under trees in the future
------------- A problem shared is a problem halved
On our last holiday in France we were surrounded by big trees, couldn't avoid them (last photo in my gallery), great shading from the heat but we had a few stormy nights including thunder and lightening, I didn't sleep, I was worried about my teens on the next pitch to us so was listening out for loads of trees potentially giving way.
Didn't have a problem with bird poo, my dog scared all the birds away (and the red squirrels).
If you camp down in southern France or Spain in the summer the first thing you will look for when you arrive on site is a pitch with trees for shade from the sun.