Yes, just like the Touch ID system on iPhones 5s-8+ and now the SE 2nd gen, the Face ID is paired to each board, if it gets broken, the only way to replace it and get Face ID is to go to an Apple store, if not, the Face ID and front camera will be rendered useless.
Yes and no. If you buy an iCloud locked iPhone, unless you know either the account email and password or the previous passcode, you will be unable to unlock it without jailbreaking it, I had this problem on an iPhone 5s.