Can the blanket Insulation in my basement be left in place when refinishing?
If the exterior of the foundation has the black waterproofing membrane, you're fine to build your framed wall right up to the vinyl blanket insulation in the basement. When I finish basements for my clients, they always ask my opinion. Here it is: I don't like spending money twice. If there is already insulation on the wall the meets the code requirement, then we'll frame right to in and leave it in place. The white vinyl insulating blanket usually has an R value of 10. In Colorado where I live and work, that's the code requirement. You won't see a lot of thermal benefit from upgrading the insulation to a R-13 or R-19. To me it's not worth the cost of removing and replacing. (not to mention the disposal of the blanket - it takes up a ton of dumpster space!) That said it would be easy to add batt insulation in addition to the current insulation after you frame the walls. 
One note: you do not want to compress the vinyl blanket with the wood framing or you'll reduce the effectiveness. If you do compress it, take a knife blade and run it down both sides of each stud and it will allow the insulation to expand into the stud cavity.

Your main source of heat loss in the basement is going to be around the  windows. Take the time to seal the windows well.  Hope that helps!