If I have a domain registered through a company like 1&1 but I want to host my files with Hostgator. I know I point the DNS entries to the Hostgator information, but my question is, if 1&1 has a server issue, will my domains then be down?
Once the domain is pointed you should be fine. The DNS servers handle the requests for your site, not the registrar.
Now, the registrar is hacked and starts repointing the DNS for all registered domains somewhere else, all bets are off. But this can happen at the DNS level too.