1) If you will live in both homes, then you don’t need to pay Khums since they will be considered exempt expenses. But if you will rent one of them to make money, then yes you’ll have to pay khums on that amount you own in the house (so the value of the house minus the mortgage loan).
The house you will be renting to someone won’t be considered an expense. So if a year passes and the house is being rented then it’s not an expense, and Khums would apply to it. However, if you had bought the house with money that you had paid khums on, then in this case no need to pay khums on the house (unless it’s value increased then you pay khums on the increase).
2) The concern is not with the rent. If you are using the entire rent money to pay for the mortgage then the Khums does not apply to to the rent money. The Khums applies to the value that you own in the house. Let’s say the house is worth $500k, and you own $200k and the rest is mortgage. When you rent this house to someone, it’s no longer considered an “expense,” so you would have to pay khums on the $200k that you own in the house. Yes, some scholars like Ayatollah Sistani have said that if the amount of the mortgage is more than the value of the house and everything else that you own, then no Khums would apply.