Very true, the only thing that can effect it is downtime. Just set up a full working version on your new server, then update the nameservers and remove the old site after they propogate.
what if IP of the new hosting is blacklisted, how do I find out? What if another site hosted on the same hosting violates the rules and gets blacklisted? would that also affect my site?
it wont do anything at all - i dont know how that IP would get blacklisted lol, then the hosting wouldnt exist now would it? just changing the domain would affect you - otherwise its all good
The IP may have negative effects on email originating from your server being flagged as spam, but SERP penalties are tied to the domain name.
It could potentially being a benefit when moving to dedicated hosting, I'd imagine google giving more trust to sites on a dedicated server due to it being a sign of commitment. You don't put shit sites on dedicated servers.