I'd say his greatest feat of strength was lifting several tons of metal off of himself when Doc Ock tried to crush/drown him. Arguably, one could also consider the time he held up the Daily Bugle on his shoulders during a fight with the Green Goblin.
Though if we count the time when his powers were augmented by Captain Universe, then you'd have to note the time he punched the Hulk so hard he sent him into orbit.
As for which villains gave him a run for his money...well, damn near all of 'em. Doc Ock was the first one to beat him. Green Goblin was the first to capture and unmask him. Hobgoblin came closer than most villains (at the time of his debut) to permenantly defeating Spider-Man, Venom was the first one that had him running scared, Carnage was an obviously severely dangerous threat, Morlun plucked his eye out and ate it before beating him to death...the list goes on.