Well that is a head scratcher for sure. How can hot water get to the cold water side of fixtures and the toilet assuming the red and blue/white pex is plumbed correctly to “all” fixtures and the hot water heater is also plumbed correctly (including any check valves in the right direction)? The only way of mixing the two I can think of is if hot and cold fixture valves are both left on somewhere. Make sure the dog wash (outside) hot and cold valves are both closed for starters as they can allow a mixing of hot and cold. Also make sure the winter bypass valves are set for summer use and the bypass leg is not leaking through. But that should be the same for using city water and fresh tank water sources. The only other thing might be that with new coaches plastic bits from coring holes in tanks gets into the water system (sometimes for years) and can migrate and plug filters; check valves and control valves. Did you ask the tech to try the city water inlet while you were in the shop?
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