Not exactly sure what you're trying to do: changing the name of the website is of course possible (but ideally it should be relevant to the domain name), changing the domain name is not.
So you want to buy say abc.com and then want to change it to xyz.com. If that is the case, answer is no you can't. But if you have purchased hosting for abc.com but want to change the hosted domain to xyz.com, yes you can do that, but you will need your hosting providers help.
Exactly. Put a 301 redirect in place though, to make sure Google isn't spotting you as a duplicate content creator and kicks your website out of the search results
Changing domain name or redirecting it to some other are two different things. You can't change domain name once you would purchase it but you can redirect it to some other domain anytime. BTW why do you need to change it?
Change domain name? Not sure what the OP meant by that, but if he bought ugly.com and wants to change it for beuty.com some registrars offer a period of grace to make change or mispelling correction, usually 7 days, but not all registrars allow to do so and who knows if the wanted name is available