Apartments need Rent

[Almagnus1]

How would the rent even be paid? Would you have to walk up to an NPC every week to pay?

Just talk to the NPC at the front counter for the apartments, and get an option that would allow you to prepay several weeks rent, and show you how far out you've paid rent.

What purpose would it serve to evict people from their apartments?

Incentive to keep paying rent.

All in all... what a terrible idea that doesnt seem to serve any purpose (more realism?).

It acts as a gil sink, and one that's a cheaper initial option (and easier to get into) than a house - which allows for a way for players to get into housing. 10k gil/week is low enough that the average player should be able to afford that for a very long length of time.