Why are light aircraft engines so noisy?
Engines could be made quieter by the simple solution of adding a muffler (or silencer as it's called in some places). They are available for many models and can be installed, at an astronomical cost. The parts and labor are actually pretty cheap, it's the certification costs that make it sting so badly. It costs a lot of money to develop and certify parts for airplanes. Mufflers aren't mandatory, and because they are so expensive to install most don't because the costs so outweigh the benefits - it costs far less to buy a decent active noise reduction headset instead. 

Propellers are a big source of noise, and there are quieter designs for many aircraft, however they have to be certified with a particular engine and model of airplane and those costs add up. 

The current certification rules are there to keep things safe, in some cases they can stifle innovation. It's why we still have carbureted engines with manual mixture controls instead of computerized fuel injection and turbonormalization. 

As for what the quietest trainer is I don't think there's a definitive answer as I don't know of a source of light airplane noise measurements, and any airplane can be used as a trainer. Of the certified light airplanes out there that are commonly used for primary training the Diamond DA20 Katana is the quietest I can think of, that's because it is much lighter than the more common Cessnas and Pipers and therefore uses a smaller engine and propeller. Smaller engine and propeller generally means less noise. Turbo-diesel powered airplanes like the Diamond DA-40s are quieter because the turbo muffles the exhaust noise.