pipes rattle when flushed, have to turn cold water on until tank fills

Couple things come to my mind here:

First, the new water heater is the very likely culprit. I would suggest you look at the temperature setting and adjust it to be slightly cooler. As the water heats up, it expands and provides the additional pressure that exaggerates the pipe rattling.  Second, the leaky toilet was probably sweating much because if the valve is leaking, the tank would get fresh supply of cold water all the time and it would never warm itself up to the room temperature to stop the sweating.

In other words, the reason for installing the mixer valve on the toilet supply might have gone away completely when the new toilet fill valve got installed. I hope he installed the slow shutdown one to eliminate the source of banging during shut-off. But in any case, the sweating might have stopped even if you only run cold water now that the fill valve is no longer leaking.   So it looks like you are trying to raise the water temperature in the toilet tank by opening the hot water valve on the mixer too much because the hot water does not start to flow immediately and you're trying to get it hotter when it eventually heats up (and hope that it's before the end of the fill cycle) so that the overall temperature is warm enough. At the same time, you're raising the heater's output temperature probably hoping that it will help you speed up the water warm-up. Both these conditions combine to create the rattling. 

I suggest that you completely close the hot water valve at the mixer and then lower the temperature setting on the heater to the U.S.  Dept. of Energy recommended economy setting of 120ºF .   The actual setting may need to be played with because there are other hot water users in the house, obviously, such as the dishwasher, not to mention humans that may like it hotter. You should also consider that you are saving 3-5% in water heating cost for every 10ºF reduction in hot water temperature.

Anyhow, like I said before, the reason for sweating may no longer be there, so try to run it on cold water only and see if sweat appears again, there is a chance that it will not. If that does not work, keep the hot water open but still lower the temperature settling on the water heater to decrease the pressure in the hot water line, it will lower or eliminate rattling.

There may be other reasons and remedies for rattling pipes, but let's start with the simplest to check first.