I am trying to understand if I chose an unreliable registra, will my site goes down when the registra is offline?

If I bought the domain, should the DNS info be reported to internic.net, then as long as it is not expired, the none cached DNS info will be on 27/7 ?

If you're using the registrar's DNS, yes.

If you're not using the registrar's DNS, no. But you won't be able
to manage it until either they're up again.

By offline, are you saying if their website is down, or is the entire
registrar itself down (i.e. out of business, legal dispute, etc.)?

usually if you using your hosting provider or your own dns it should not be down

unless if the registrar us down and all domains goes on-hold then name servers delegation will be suspended i believe.