I've ceiling mounted my projector. I would not want it just in drywall. Yo definitely want to find a joist.
For the hot water issue, you can get a temperature balancing faucet for your shower/bath. Life changing. They aren't even ridiculously expensive. I'm not sure if you can just swap the internals, or if you need a whole new unit, though. You can also set a limiter on it, so it doesn't go to hot.
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