Domain name is like your name, when someone needs you they call out your name, much like online. The internet is a bunch of address (IPs) and as it turns out humans do not easily save numbers (I dare you to re-call your dad's phone number from memory) so domain names invented to convert IPs to names so next time you call esmailelbob.xyz you actually call its IP so yup, You can live online with your VPS's IP, but it will be hard for your users to visit your site each time, let alone reverse proxy (a topic will be discussed later) so it's better to buy domain name and because you are lucky, It's already cheap from places like Namecheap or GoDaddy, but I recommend Njala.
Well, it depends on your end goal, if you want to be censored for example try to choose google domains or GoDaddy.
Offshore domain names or resellers are companies who buy domains in their name. You see normally when you sign up for a domain. You need to type your personal information like address or real name and then later sell you a "whois protection package" because anyone can use tool called whois to look up domain information so yes your personal data is public. but fear not! Because offshore buy domains in their names, your data is actually protected. There are so many offshore domain resellers, but I personally use Njala because you can sign up using XMPP and you can pay in crypto so in theory you can be complete anonymous even from Njala themselves.