Yes. Don't know. No. Yes. Until graduation. Assume as long as player does not sign with an agent, he is okay. If the player has a good year, NHL team will try to sign him at end of NCAA season.
Eligibility: Any North American player who turns 18 by September 15 and does not turn 20 by Decemeber 31 is eligible for the draft. Any non-North American player can be drafted at any age if they are at least 18 by September 15. If a player is 18, they must declare themselves eligible, but if a player turns 19 by September 15th, they are automatically eligible for selection in the draft.
Signability: If a player is not drafted by age of 20, that player becomes an unrestricted free agent. A player not signed by drafted team within two years can reenter draft, assuming they are still eligible, and if they are not eligible, will become unrestricted an free agent. A team has the rights of an NCAA player until 30 days after the player leaves the college. If a player is drafted a second time, they can not reenter the draft. If a player has entered the draft twice and not been selected, they then become free agents regardless of age. A non-North American player can not be signed unless they are drafted first.
https://www.draftsite.com/nhl/rules/