Once registered each Minecraft account has it's own name and UUID. While Minecraft names can be changed every 30 days, UUID of each Minecraft account is permanent and can not be changed. After changing your Minecraft name, your old name will be locked for 37 days, giving you a chance to get your Name back if you want, after 37 days anyone can take your old Name.
When changing Minecraft names you should be careful not to pick explicit or offensive username as you will be stuck with it for 30 days and most of Minecraft servers will of course ban players with such Names. On most of Minecraft servers your data is tied to your UUID not your Minecraft name, meaning you should keep all your progress but of course that could vary from server to server. This also means you can not avoid bans on Minecraft servers by changing your Minecraft name.
There are some old 2 character Minecraft Names ,but nowadays each Minecraft name has to consist of 3 to 16 characters, only letters, numbers and underscores are allowed. If you want to know if specific Minecraft name is available you can use official Minecraft page or our Minecraft name checker.