The issue I am having is that very often, the water in the cold line from my taps is cold for a few seconds when I turn it on, and then it warms up almost immediately and stays warm.
Last year, I was having the same symptoms. I had the plumber out (builder's warranty was still in place), who discovered hot water was flowing backward through the cold water inlet of the hot water heater. He said the backflow preventer in it must have failed, so he installed a check valve on the cold water inlet right past where the expansion tank and shutoff valve are.
Fast forward to a couple of months ago, and the problem has been slowly coming back again. Now it's worse than it ever was. Sometimes the water seems cold and stays that way, but more often than not, it starts cold and then gets warmer.
Initially, I thought the check valve the plumber installed must not be working. There is a shutoff valve on the cold water inlet right above the expansion tank and below the check valve, so as a test, I turned that off completely, and I turned off power to the water heater.
Then I went around testing faucets. As soon as I opened the bathtub mixing valve just a little bit (trying to get only cold water to come out), the water got really warm. I let it run for a bit, and it stayed warm. I turned it so that barely any water was coming out, and it still stayed warm. Then I noticed the cold taps nearby all got warm quickly and seemed to stay warm.
Does this mean I have a bad mixing valve at the tub faucet? I'm going to try and shut off the hot side if it's possible to do so at the faucet so that I can see if the problem stops.
If it does turn out that's the issue, it seems strange that my house (built in 2020) would run into two seemingly unrelated plumbing failures that both caused the same behavior (hot and cold water mixing). Could that possibly be a sign that there is too much hot water pressure or some other larger issue, or am I just reading too much into two separate events?
Update: I forgot to mention I was doing a load of laundry on the tap-cold setting on my washer at the time. After re-testing everything, this time the tub faucet stayed cold when it was the only thing on, but as soon as I turned on the washing machine to cold, warm water started coming out of the washing machine as well as nearby faucets.
I'm thinking the bathtub water was warm because of a possible washing machine issue, so I'm attempting to leave the hot water shut off at the washing machine and see if the problem is affected in any way.