Two different things
The domain is the address; the hosting is the building the site sits in. You can move hosting without changing your domain, and vice versa.
No — you keep the same domain and just repoint it at the new host. The address stays; only the building changes.
Keep them separate and in your name
Buying them from the same company is fine, but make sure both accounts are yours. If you ever change providers, owning both means you're never held hostage.
Key takeaway
Own both the domain and the hosting accounts, and you can never be locked out.