Aerator has nothing to do with the dishwasher. It should be fed from the hot side. If debris goes into the heater, it will settle to the bottom. Definitely check the aerator for the sink faucet but you may need to clean out the stops under the sink.
Typical install is $1500 including new heater and disposal of old one. Will be more for a thermal expansion tank, copper pipe, t&P line, vent work or extra gas fittings (bringing up to code).
The 3.7’s are very reliable engines. Been in service for a long time. We have a 2018 Q50 with the twin turbo 3.0. No issues with ours but it’s known to have a host of issues with turbos and accessory belt.