Nobody is exactly sure how much change will happen how fast - it may depend a lot on what we do but even then weather systems are highly susceptiple to tiny changes which can amplify quickly
I think that even the most pessimist predictions don't have the Gulf Stream shutting down for 100 years or more.
Even if the gulf stream were to shut down it wouldn't necessarily be an ice age, after all we are on a lattitude similar to Nova Scotia and that's not under Ice sheets - but it would get somewhat parky!
Sea level rise again may be difficult to be sure about, in the last 100 years it's globally only been about 20cm. The general feeling seems to be worldwide it's likely to be about a meter over the next 100 years.
The problem is that this could be much bigger extreme events, when tides and weather combine there is more water in the system to slosh about and get over sea walls. Bad but images of half of the UK being permanently submerged are a long way off.
I'm not sure why you equate rising sea levels with a new ice age though