You should be OK to keep the vents closed in Winter. HOWEVER, you SHOULD buy some standard pipe insulation at the Home Center. Someone needs to determine what piping/plumbing you have under there and then insulate it. If you and/or your significant other are handy enough, you can figure it all out yourselves.
Yes, in a sense, you can waterproof a crawlspace. However, it's a bit costly after-the-fact. The best way is to install a French Drain / Tile Drainage system around the perimeter of the ENTIRE house and then divert the water AWAY from the house. With the type of problem you're describing, the application of something like Drylok Sealer won't be enough.
Start by taking a 'look' at some relatively simple measure that might actually solve, or dramatically alleviate, your problem(s). See my FOllow-ups in Leaking Basement. The same info can apply to your situation. If you need more info, post back or e-mail me directly.