1. Yes, absolutely. It would be due to a change in supply. Price controls, taxes, regulation, etc affect the price only insomuch as they affect supply and demand. It's a matter of definition: it's just as true in a market economy as it is in a non-market economy, or under market failure. Whether it's a useful framework is a different matter!
2a. So, interest rates are indeed determined by the market, with respect to open market operations right? The Fed buys and sells with a target in mind, but they certainly can, as you point out, fail to hit their target.
2b. This was a bit of unfortunate jargon: demand for "money" specifically means cash/cash equivalents relative to other assets like stocks/gold/real estate. It's not referring to wealth, although I do think the desire there isn't infinite: biological limits/post scarcity.
2c. And "demand" for money refers to the whole of the demand curve (and shifts in the whole of the curve). That is, how much of your portfolio do you want to hold in cash/equivalents at a given interest rate?
2d. Fed open market operations do target the whole of the economy: it's a ham fisted approach. Inflation affects groceries and gas, just as much as bank liquidity and such.
3. So, relative price change isn't inflation by definition, and market expectations of the interest rate path are priced in. Of course, there are plenty of relative price changes alongside inflation, but there's a lot more to the economy beyond open market operations. The Fed isn't able to specifically target say apples over oranges, hence ham fisted.
4. Sure, broad aggregates like NGDP don't include inequality or climate change, but that's a good thing because the Fed's job is limited to keeping aggregate demand (NGDP) on track. Of course there's a lot more to the economy, but that's beyond the realm of Fed open market operations. That is, I don't think the Fed is the right place to look for solutions.
5a. So money neutrality only says: a permanent 1 time increase to the money supply doesn't affect real GDP.
5b. Again, more jargon: "long run" is however long it takes for thing(s) to adjust (whatever it is: wages, rates, output, etc), and "short run" is just short of that. So as a unit of time, it depends on the good as well as real world conditions. The short run can go forever, market failure maybe. The short run can be zero, when things go according to market expectations and everything's already been priced in.
Sorry, a lot of that's non sequitur. But that's my point: I think the mainstream is a coherent and useful story, but it's not relevant in this context, hence total nonsense. To the extent that there's an upward trend in inefficient businesses or 'rich get richer, poor get poorer', I don't see how open market operations can be the culprit.
But yeah, I don't necessarily disagree with your overall impression of the Fed hahaha.