You can log in with OAuth, or by Email.
You can change your _display name_ on your profile page, and that's what appears publicly on the site (your email doesn't appear publicly, even if you log in by address!).
Today to get a custom profile URL like radiopaper.com/Dave, you just need to ask and we'll create an alias for you. There isn't a self-serve alias mechanism available today, but we're thinking about how to provide one. Any profile can be linked by the URL radiopaper.com/uid, for instnace my profile can be reached at https://radiopaper.com/user/AwQLwnOgQFdyDfNoYfNCSSsVdx43, "Dave" is just an alias.
Clearly having separate concepts of login method vs display name vs profile URL is all a bit confusing, this is helpful feedback!
Also, having a section in the profile where I could change my login email address would help with the UX.
Being able to change your email address is a known gap, sorry about that. For now if you want to use a different email you'll need to make a second account. We plan to make account merging possible in the future.
A short-sighted technical decision on my part: we use the Firebase Auth UIDs as our internal user IDs, but Firebase Auth does not allow you to have multiple email auth-providers on a single account. So we need to add a layer of indirection in our data model.