Is it safe to sleep with earplugs everyday?

Yes, with care. Ear plugs are generally safe however they may push wax into your ears and cause a cerumen impaction in some folks. Also keep them clean or replace them regularly to prevent contamination and possible infection. If your wearing them due to painful hearing, and not just to reduce noise so you can sleep better, your should see a doctor to be evaluated.
Yes. As long as they are not pushed so far in you can not retrieve them and keep them clean!