When someone becomes banned from Lemmy, do all their posts and comments automatically disappear, or does the username just come up as invalid?

It depends. When banning someone there's a checkbox to also remove all their content.

If they are not a premium Lemmy user, unfortunately all their property is liquidated, followed by their entire genetic line, then all their content is deleted character by character slowly over 30 days time. Not a premium user? Subscribe today!

No. You can still see profiles and posts of banned users unless they’re otherwise removed. The profiles say “banned” in red at the top.

That really only applies to the server they were banned from. The network aspect of federation complicates this a bit.