2) I am unclear why one would cap the number of houses or prevent rental. Or how that would solve the problem. Affordablity is only one reason people rent. There are alot of people that do not WANT to own. Hell I am a home owner and have often thought about selling and going back to rental. I might if I ever need to move cities again
3) Price fixing and having government take over the housing market is not a solution to what is largely a government created problem in the first place. Zoning, non-saftey based building standards (i.e min square footage, min lot size, bans on Multifamily, etc) and other government actions are the direct cause of the housing shortage. Change Zoning Laws and allow builders to only have to adhere to actual safety based building standards and housing shortage would be over.
Government putting its hands on the scales of the market for over 50 years is the root cause of the problem, everything from FHA to Fannie/Freddie, to Zoning laws all created this problem.