I really can't speak for the gnats, sorry.
As for mosquitoes, we have a 'stream' that runs by the house which is really a stagnant ditch and a breeding ground for all sorts of insects with aquatic larvae. My nemesis are Deerflies. I would take all sorts of mosquitoes over Deerflies. If you have any luck with those I am all ears!
As mentioned before, a cornerstone of mosquito control is management of all stagnant water (I can't control mine because of the stream). There are larvicides you can use that have a bacterium
Bacillus thuringiensis that will kill off the larvae. You can use these in ponds and such. I have horse tanks and just put guppies and goldfish in the tanks and and they eat the larvae as they are hatched. The guppies will not survive my cold winter, but the goldfish will.
The Mosquito Magnet I have heard is really good, but is out of my price range.
I used to have a bug zapper I would turn on at night and it really worked very well, if you can get used to the zzapping. Our neighbors have one outside their bedroom that I can hear 200 feet away. Some folks complain they are non-selective but honestly there were very few other insects at the base of the unit, just a big old pile of dead mosquitoes and a few moths.
We have a large number of barn swallows that I have encouraged to breed and multiply and they have eaten so many mosquitoes I didn't get a new lamp for the zzapper when it went bad and we are really doing quite well without it so far.
As for DE, it will only work if there is direct contact with the insect exoskeleton such as grain beetles. I personally tried it for mite control and it did not work at all for me for that application. I know some folks even feed it to animals and to themselves, but it is really hearsay and I have found no studies to support its use as an internal parasiticide.
I have a very organic friend that uses concentrated vinegar sprays (via a 2 gallon sprayer) on any plants she wants to get rid of. It requires repeated application but is very safe, if you don't mind your yard smelling like a salad