What is a character that another player introduced that they were playing that either instantly or eventually began to show some red flags for anything from being a disruptive player to full main character syndrome?

A backstory that includes all kinds of bullshit they did prior to becoming a 1st level adventurer. Saving the kingdom? Town? Village? Really? Don’t you think you’d be more experienced than level 1 then?
Large backstories in general.

Anything with a gun (that isn't the artillerist (most of the time), or in a setting where they're common), or by extension, a player who really want their characters to have one.

How come? I'm pretty happy to incorporate guns into the setting if a player wishes to use them.

A character created with deeply-held beliefs that put them at odds with other player characters.

The worst example I ever saw was a journalist in a World of Darkness game who was determined to record and expose all the supernatural goings-on to the world at large. (If you're unfamiliar with WoD, one of the core conceits is that you don't let the normies find out about fantastic stuff. It has very serious consequences.)

Insisting on having a +3 item.
Yeah, you can explain it 10 times why it is in your BG, but no, it is not ok for a LVL 4 party

"I saved the town of a horde of Orcs, escorted the maiden fair through the Feywilds and defeated the Lich King of Azmerog!"

"Bro, you have a set of studded leather and a shortbow with normal arrows."

"I LISTENED TO YOUR SHITTY BACKGROUND STORY WITHOUT INTERRUPTING, DID I NOT?!"

Ba-dum-tshh

Lone wolfs, too obvious anime references, race+class combination for the sake of build instead of story

Sometimes I seen people who wants to play in non-eastern focused game eastern characters with katanas and everything. They never was executed good.

"He sits in a dark corner of the tavern. Alone. But like mysterious alone. And cool. He doesnt intract with anyone, but you feel how he observes everything and notices everything. He's cool. Like dark cool. And mysterious."

Having a level 10+ backstory while starting at level 1.

Bonus points if said character has nothing to do with the world nor the narrative.

Chaotic evil Drow rogue.

World building implications aside...Read the content of Ops post.

A character named Edge with the Noble background.