Signal could allow both options: register with a phone number or with completely anonymous logins. I think they just don't want anonymous users, visits from FBI etc. Also, it is not very profitable to sell ads to anonymous users.

You have to have a secure channel to exchange phone numbers too. The code could be communicated simply using a QR code. Other messengers already work this way and it works well.

Yes, the difference being that a phone number is an established identity mechanism and you likely already know it.

Yes but it's a kind of bad one. There are many ways you can lose or change your phone number. Email address is a better identifier.