Yeah, BT is great stuff -- among the narrowest-spectrum, non-persistent pesticides in existence (it's actually a live bacteria). Make sure you use BtK ("Thuricide" -- the K is "var Kurstaki"), as that's the version that targets caterpillars. Different varieties of BT target different types of insects. BtI (var Israelensis), for example, targets mosquitoes and flies. Completely nontoxic for humans, birds, fish, etc. You can even drink the stuff. The main downside is that it doesn't kill them instantly (although they do stop eating well before they die) and needs to be reapplied every few days until the buggers are gone.

Thanks. so should i add some dry shredded paper?

Both the flies and mushrooms point to one problem, excess moisture in the mix. The mushrooms are an indication that you have an active fungus colony in your compost, a normal part of the digestion of compost process. Those fungi need a fairly moist environment to produce the seed stalks, the mushrooms, that will spread the colony around. The same thing happens in many lawns when that soil gets an ample supply of water. The flies are looking for some place to lay eggs and one of the places they look for is decaying organic matter that is quite moist and warm. Some tumblers have provisions for getting rain inside, while the air holes of others can allow rain inside, and that can cause the material to get too wet. Add enough dry material to absorb the excess moisture.

I have a rock mulch in the Kaffir lime pot, so that's not an issue. But these cats like their greens. Bringing in wild mallows and mustards for them helps, but they will eat plants down to the nubbins if given a chance.