I’ve been on Safari with my iPad for years on this site. It works fine with Safari. You probably need to check the “remember me” box when logging in next time to remain logged in each time OR if you refresh your memory cache or clear your web “history” on your device, that might also log you out. I also just tried clicking on the link while on Safari on my iPad. Opened the link with me logged in. Not sure if any of this is helpful, but hopefully.
As for changing your user name, sounds like that has been covered above.