There are organizations that are called registries that manage each TLD option. When you register domain names, you would first visit a domain registrar who would in turn check with the registry that manages that TLD to see if the domain is available. If the domain for the TLD you want discovers that it is in fact available, the registry adds it to their registry and gives you access to it. If the domain has already been registered by another user for that TLD, you will be shown during the domain registration process that the domain isn't available.
Absolutely. Owning 'yourname.com' (as well as related domains such as yourname.tech or yourname.me) is a great way to brand yourself and retain control over your name's online presence. With a personal domain name, you can set up a portfolio, blog, or hobby site that’s associated with your own personal life. You can also set up a custom email address like email@example.com allowing your visitors an easy and memorable way to reach you. So even if you have no immediate use for yourname.com, it's wise to register a personal domain to ensure that you (and not some stranger) control your name online.
1and1 have terrible customer service and billing, quite possibly the worst ever encountered. Their billing/invoice system is all over the place, their staff untrained and unwilling to resolve and their pricing is literally plucked out of thin air. Beware, pay 12 months in advance (not monthly) then cancel the direct debit or Paypal auto payment, otherwise they will rinse your pockets when you’re not looking &/or unexpectedly take an extra payment because of an ‘oversight’.