You don't need the stove. You normally need to dig the stove in to get the best out of it as well. Carry 6 inch nails and a small foil tray. Dig little hole, place in foil tray, hexi tablet is placed on tray, place 6 inch nails in ground to support pan just above ground height. If you wish to reduce weight, use 4 inch nails and only three of them.
Hexi is inefficient if it's not given a good windshield / windbreak, that's one of the reasons why the military used to teach to dig it in. When cooking with it make sure that whatever your heating vessel is that it has a lid, you'll get better performance.
Cleaning, yeah wet sand, wet grass, teabag, pretty much any wet item with a mildly abrasive action will work and yes as soon as possible after having cooked with it. The smell is toxic in confined spaces so you can't really use it in the awning of a tent if the awning is all closed up, it's common sense, just ensure you have reasonable ventilation.
It's not the quickest of fuels, it's not the cleanest of fuels, but it works in pretty much any environment without the requirement to do anything to it, you don't need to learn how to use it and it's simple. It is in many ways the ideal survival fuel. On the mountain and arctic side of life we used to be taught that every man carried two tablets of Hexi along with his waxes and his candles, they were carried in your jacket.
I still have some in an emergency pack that I carry when out and about.