is there a quicker way to find archived emails in gmail?

quickest route if you roughly remember it: just search. gmail search covers archived mail as well, archiving only takes the inbox label off.

archive isn't a folder, that's the bit that trips everyone up - the mail just leaves the inbox and stays in All Mail. tap the menu top left and scroll down to All Mail. if you want only the archived ones and not the whole pile, search -in:inbox -in:trash -in:spam and it'll show just those. the sign-in prompt sounds like a second account on the phone rather than anything to do with archiving.