Retirement tax questions

Follow up question 1: Can someone double confirm that her payment on property tax would not be considered as income?

Follow up question 2: Some earlier posts have suggested that if it would be in the form of rent above and beyond sharing household expenses, it would be considered as income (e.g. paying a fair market rent for a room comparable to those advertised on Airbnb in the same township), is this correct?

Final question: If a few years down the road, she ended up in nursing home and had to apply for Medicaid, would any of these household sharing payments be considered as gifts thus subject to Medicaid look back penalty?