No. That is not a problem. Please read about the Slasher proof of stake algorithm. It is very easy to punish the creator of many possible blocks. People always use this nonesense phrase that "with proof of stake nothing is at stake". That is no problem. If the protocol allows that your mining profit is taken away and given to the next honest stakeholder then why do you say that it is a problem that nothing is at stake.
Yes it is a problem and last I checked, there's still no working PoS implementation that doesn't rely on central check-pointing. "Slasher" is an idea (likely full of unforeseen issues), not an implementation and until there's a working implementation that doesn't rely on central check-pointing, PoS is vapor-ware.
No, it's very easy to convince yourself that these problems are solved when they in fact are not. I have not seen any proof-of-stake mechanism which does not either require centralization (despite claims to the contrary), or devolve into proof-of-work as miners grind blocks until they find one that names themselves or one of their sock puppets as the next signer.
that part of the problem is partially solved. not every node has to "mine" all blocks. in fact in PoS there's no mining at all. the big problem is that you create all coins at genesis. there are a bunch of proposals and work in this direction.
It sounds like it should be, but it isn't, lacking PoW, there's no reason not to create many possible blocks and without central check-pointing the whole thing falls apart. The problem with proof of stake is that nothing is at stake.