You are correct. You do not have to report alimony if the divorce was done after 2018. If it was prior to 2018, then you would need to report it as income.
If you are NOT paying more than half of the cost to maintain the home you live in with your child, then yes, it could affect it if your ex-husband is paying the mortgage. Utilities, maintenance, insurance, etc are also part of the cost of maintaining the home. However, if he is covering the entire cost of all of that and you are living there with your child, you are not covering over half of the cost of maintaining the home.
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