There is indeed still space.
There is no real limit on archetype due to being outside of established yugioh canon, but if you can make your archetype chosen fit your themee. E.g Gem knights from a mining family. That would be preferred yes.
for the other question, Yes card rarity should be sort of considered if by rarity you mean "Is this card a one of a card, or did I sacrifice my wife to the god in order to unlock this power.". In a way that some of your top end boss monsters would be better if they were earned. E.g lets imagine I was XYZ (the fusion monster series), Later on I'd unlock VW, and then later on from that through perhaps a really epic battle against it I'd unlock ABC,
Before finally culminating in the most epic summoning of A to Z
Generally speaking, if you feel like your story would benefit from you gaining cards then yes you can asssume card rarity to be a thing.
Limits to character backgrounds are minimal, As long as it isn't something mary sueish, You can be a washed up castaway on an island that grew up with nothing but trading cards and practicied his card draws in the woods until he was rescued if you so want (don't do this, this is just taking it to the logical extreme). But you can't be a Seto Kaiba (at least not ot the level of the actual seto kaiba) if you catch my drift.