The Moka pot is an excellent brewing tool, it's reasonably inexpensive and foolproof, and can be used on virtually any heating equipment from gas hob to electric to induction to camp stove or even open-pit fire (though you'll want to ensure the handle doesn't melt). Results are better than appliances costing 10x to 20x more, at far less hassle.
Another reason for taking the pot off-heat quickly is that the rubber sealing gasket will last far longer. Overheating degrades those quickly, and burnt-rubber-tasting coffee is not fun.
I've never had any particular problem with cleaning with suds and sponge, but a periodic soak in hot water with baking soda will remove coffee residue and tanins. (Both are acidic, the soda neutralises and removes it.) Stainless steel handles this somewhat better than aluminium (the latter may dull through oxidation, though so long as your soak and rinse are reasonably short that's not a major consern.) There are coffee cup/pot cleaning solutions which might also work.
Cleaning both the metal filter plate and the inside of the riser tube on the top part of the pot also help. An old toothbrush is useful for reaching into the latter. Smaller is better.
The baking soda trick also works a wonder on mugs and tea-stained pots or cups. Let it stand 10--30 minutes. I usually add a drop of washing-up liquid as well. Sponge or brush afterwards, the stain should lift straight out. Rinse & dry.
