I signed up a few days back all proper, verified and all, but my username still shows up as "user21414". Is this because I need a certain reputation?
When you first create an account, the default display name is the word "user" followed by the unique ID on the site where the account is created.
In order to change this, you must edit your profile. However, the android app does not support editing the profile. There is already a pending feature request asking to add the ability to edit profile via the app, but since the app is currently unsupported, it's extremely unlikely we'll see that feature implemented.
Bottom line: if you want a different display name, you must use a web browser, and if on mobile, make sure to go to "full site" mode, there is a link in the footer, since the mobile view of the website also does not support editing the profile and has no link to the edit page. (and yet again, there is a pending feature request asking to add such ability.)