You can request a UserID change by contacting customer support. You will be required to prove ownership of the account (security question, etc) and you will be asked to provide the name you wish to change to.

You can check availability of UserIDs from the account creation page, just ignore the other fields and type your desired name into the UserID section and then click 'Check Availability'. This will save Zenimax from having to waste time responding to you something like 'We're sorry but the name you have requested is already in use'. It will save yourself time as well.

If you unfriend somebody prior to having your UserID, then no, they will not be aware that your handle has changed. Unless of course they remember your character name and thus use that to figure out your new UserID.

(Friending, mailing, ignoring, etc all reveal UserID even when enacted against/with a character name).