I feel dirty even getting into this, but since people seem to be having a hard time with Cap pulling him, this is no different than a 200 lb. guy trying to lat pull down over 200 lbs. Unless he braces himself under something, no matter how strong he is simple physics won't let him pull more than his body weight down. Same thing with Cap pulling Spider-Man off his feet. Cap clearly outweighs Spider-Man, so the only way massively superior strength helps in that situation is if you can brace yourself properly. Spider-Man would have had to use his wall crawling feet on the ground to counter that move, and it isn't that big of a stretch to assume he wasn't ready for it (spidey sense shouldn't make a bit of difference, since it is clear at that point Cap is already a threat).
Also, it's a comic book movie. Nothing that happened was too hard for me to believe from a Spider-Man that is still getting used to recognizing and using his powers. Just because your spidey sense tingles doesn't mean you instantly know how to counter the threat without some practice.
Also also, just because Peter Parker has been an adult for the majority of his comic run doesn't mean he has to be in the movies. He's been an "adult" with mixed results in multiple movies, why not give this one a shot? Not to mention that technically this kid is practically the same age as what the other versions were supposed to be too, it's just that he actually is a kid compared to the 25+ aged actors they got to play him before.
