Just wonder what if my mail server went offline for some periods, and the sending party couldn’t deliver.
Will there be any consequences except I don’t get the mail? I tried searching but they all in the perspective of a sender and get a bounce, rather the other way around.
if you need more reliability you can always use a second mailserver as backup, add a second mx record to your domain and if the first one is not responding the second one will be used. there is no limit of how many next servers there can be specified afaik.
Sure. But you could also just not turn off the mail server for a start ;)