Not every dishwasher has a heating element. My dishwasher does, and the manual still says to run the hot tap to make the dishwasher more effective.

Also, just because your dishwasher might have a heating element, that doesn’t mean you always wanna use it. You know what the “sanitize” setting does if you accidentally leave some oily residue or food on your dishes? Everything gets some junk baked onto it. And then you (actually, me) have to scrub at hard, dried crud that you should have handled before loading the dishwasher the first time, but you were lazy and let your wife wash the dishes, and you didn’t know at the time that she doesn’t inspect things before baking them in the soapy dish-steamer. (Rant over)

As for washing your hands while the dish washer is running, I’ve never heard of this issue. Most residential plumbing is capable of carrying somewhere around 6 GPM (Gallons Per Minute), and your faucet is probably limited to 1.5 - 2.2 GPM, but if you wash your hands at that pressure, your shirt is probably gonna be soaked from splashing or spray. Your dishwasher only takes a minute or two, maximum, to fill, and the remaining 3.8 GPM is more than enough. Once your dishwasher fills, the pump (sometimes heated, sometimes not) cycles that water through the jets, and then eventually pumps the water out the drain hose.
If she’s upset because you don’t like using scalding-hot water to wash your hands, then that’s also a non-issue. Your hot water lines are connected to your water heater, and the hot water lines will deliver water from your heater whether you want them to or not. Your cold water and hot water lines shouldn’t interact behind the faucet. You can run all the cold water you want at your faucet and the only difference is that not only will the hot water supply line be hot, the cold water supply line will be cold. Separately.

Now, something you shouldn’t do is run your disposal at the same time as your dishwasher. When you install your dishwasher, it specifies that you should install a “high loop.” This just means to make your dishwasher’s drain hose go as high as possible under the cabinet before reaching the sink drain. It doesn’t have to be a literal loop, but I like making an actual loop because I like the way it looks. This “high loop” acts as a vacuum break to reduce the chances of your dishwasher sucking dirty water back into it. If you recall how to make a siphon, the two things you need are an unbroken hose with no air pockets to disturb the water flow, and for the source to be higher than the destination. Your dishwasher’s drain is at the bottom, close to the floor, and the hose intersects the sink’s tail piece around knee-height. The high loop makes it more difficult for backflow water to fill the cavity of the drain hose, so you’d pretty much have to clog your sink drain past the p-trap, and then fill the sink to nearly overflowing for you to move water from the sink drain to the dishwasher. BUT! This isn’t true with a disposal. Disposals are capable of generating enough pressure that they can actually push some water up the dishwasher’s drain hose. So, if your dishwasher is draining while you use the disposal, and it stops draining while you’re still using the disposal, you might actually be able to get some of the ground up food from your disposal past the high loop. Even worse, your disposal can often let through food particles that are bigger than the jet holes in your dishwasher. Since your dishwasher fills and then pumps the water already inside it to spray your dishes, you can end up putting food particles through your pump, breaking it, or through the jets, clogging them. Also, you’re taking that dank smell from your disposal and using it to clean your food.