The owner equivalent rent measure that the Fed uses in the US has been decoupled from Housing prices for a long time. House price increases or rent increases don't necessarily have an impact on inflation if few are paying the marginal rate, and they choose to eat the higher cost rather than asking for more money.

When a critical mass of individuals pay the marginal rate for housing, and they choose to demand more for their services to compensate - then it will show up in the inflation reports. I'd argue that it was a miss for the Fed to focus on owner equivalent rent rather than a broader measure of what consumers are paying for housing as the overall mix of housing has also been changing as the number of investment properties increases.

If your monthly payment stays the same and the interest rate is lowered you pay less interest to the bank which means the seller gets your money instead. The price increase literally doesn't matter to you.

This depends. Housing cost if you're considering the price of owning a home is not part of CPI for the same reason that stocks are not a part of CPI, that being they are considered assets.

If you talking about specifically renting housing, then correct(in US).

That depends. Here in NL, housing costs are explicitly excluded from the official inflation numbers, presumably because housing is still seen as an investment rather than a short.