Needs not, but I don't see why not, if they want it to. I'm not a big fan of the Raimi films, but I did like how they had a personal connection, especially in the end. Personal preference, I suppose.
Lizard is an accident true. Electro, I don't know. But the others seem to be intentional creations of Oscorp. I think that's an improvement personally.
Again, with Doctor Octopus, I think it's premature. We don't know the circumstances. If he works for Oscorp, then it stands to reason that he would leave his arms at work when he goes home (assuming he hasn't become a super villain yet). Maybe they're a prototype he developed there. We just don't know yet.