Thinking this over a bit *may* be that you have a small tank with a high volume output pump and so the pump may only need to run a few seconds at a crack.

However, if you're getting an odor from the tank ...something isn't right. It may be nothing more than the lack of sealant or a gasket for the lid.

Impossible to say from here what the problem(s) might be, but sure sounds like something is not quite right.

The short pumping cycles *might* also be related to the float switch having come loose from its intended attachment point or similar. If/when you open things up, do some "fill and pump" cycles withthe tank open so you can see what physically goes on in there.

A missing gasket can usually be replaced with a bead of silicone caulk in which the lid is then bedded. Although this is not he handiest arrangement for future entry, it will seal the tank lid. An alternative would be to lay the bead of silicone and then quickly place some pieces of saran-wrap over the caulk, then gently bed the tank lid on that. Once the silicone is fully set up, you should be able to lift the lid, remove the saran-wrap and then seat the lid directly on the caulk. Then place the perimeter screws to tighten things down.

Can't help but wonder if the vent pipe is properly vented and migth be the source of the foul odors. Likely is, but one never knows for sure. I've seen some very goofy shortcuts taken which couldn't possible work right...and didn't.

I suppose there is also a chance that one of the associated DWV pipes has a crack or hole in it from an errant screw. All sorts of things could cause the problem odors you're experiencing.