I don't think he'd work in the DC Universe. He was part of what changed it. The idea of a boy (or at least teenager) as the superhero who fought authority figures was kind of a reversal of DC's image of strong manly men as the heroes and boys as their ambiguously creepy-underage sidekicks.
I don't know if the squared jaw, whit bread-ness of DC pre-1970s would have allowed Spidey to work there. If he was there, he wouldn't be the flagship and he probably wouldn't be as popular as he is today (which is moreso than any other DC character, save maybe for Batman). I think Spidey is too much of a Marvel product which is an inversion of DC's original formula.
If he was at DC, he'd probably not crack jokes, would be over 30, deadly serious, have a lot of cosmic adventures and have passed his mantle to at least three other people at various points.