All members of the kin should be able to pay the upkeep. You can pay up to 6 weeks in advance (a month and a half). If you are in your final week and have not paid you will get the email notice (only the leader of the kin).
If the rent isn't paid by the expiration date on the rent (which all kin members can see just by looking in their character panel under housing) then the house locks. It does not foreclose. We used to foreclose on houses. Foreclosure means it was taken away and you'd have to buy a new house.
If the leader has not logged in within two weeks of the mail being sent then it will delete (like all mail).