Picnic Tables: Yes. Some of them are even in the shade. This is not a given, by the way. I'm not sure that the people who plant the trees and the people who park the tables ever talk with one another at all, or look at how far shade is likely to reach.
Trees: It has plenty of trees. I counted at least four kinds, most kept to the tall, straight ideal, not my favorite kind, I admit, but a tree is a tree, and I'd rather straight than none. They also included a couple of types I had not seen in other parks, so hurray for keeping things interesting and varied!
Two were even flowering when I visited. It's not the sort of park that really invites sitting down and pondering the nature of the universe, but still, the trees are there to provide grace notes and to shade the path.
I did not do a very detailed bug-hunt. Nothing notable flew up when I scuffed my feet. I did see a Phoebe sitting on one of the picnic tables, though, which is an indication of at least some bug-ness. Overall, I'd put the critter count at higher than average for the city in general, but I wouldn't suggest going to Langenbeck Park to bird watch.
The landscaping is not especially welcoming for birds and beasts, being mostly clipped grass and the aforementioned trees. No raggedy edges or ripened grass for anything to hide in or eat from.
Drinking fountains: Also no. (As someone commented, "If you don't drink any water, you won't need to pee.")
Sports stuff: No. No kite flying, either, for obvious reasons.
Parking: On the street. It looked like there were plenty of different neighborhood areas with access to the park, so I doubt there's much trouble finding space, though I do advise checking street-sweeping days and the like.
Play Structures: I can't remember, and I didn't make notes on this. I think not, but I'm not going to swear to that.
Will you be able to forget you're in the city? Not with those towers and that hum overhead. That said, there are worse uses to make of power line land (I've seen some stretches of rocky, barren wasteland, all fenced off and good for no one, not even the bugs). Also, there is a nice, long, curving sidewalk running the length of the park, and I imagine that it must make a pleasant addition to the daily walk or run. I saw quite a few people walking their dogs and one person intently rollerblading along the way. It looks like it might be nice for varying a bike route, as well, but I did not think to check what the park regulations said about that.
As an added bonus, someone who wants more than half-mile Langenbeck provides can also cross Magnolia at the crosswalk by the nearby Home Depot and continue on through Talbert Park.