Should there be standing water in this part of the dishwasher

Modern water saving appliances seem to be prone to this. Usually using proper detergent keeps this in check. Since the smell goes away after a wash, it sounds like you are doing this. You might try a different detergent anyway. It may also be a plumbing issue if standing water is consistently left at the end of the cycle.

Unless you're on a septic tank, start a empty wash cycle and let it fill with water. As soon as the water starts circulating, add 1 cup of household liquid bleach. Use 1 pint white vinegar if on a septic tank. Close the door and let the water circulate for 10 seconds, then stop the cycle and let it sit for 1 hour. Restart and finish the cycle.

Clean the filter and screen with hot soapy water. This should get rid of the smell, but it will eventually return.

Clean the filter and screen on a regular basis, perhaps weekly. When the smell returns, redo the bleach treatment.

It looks like mold or black mildew.

It commonly starts growing on and in the door gasket, which is somewhat porous. It collects on the filters until, as you have learned, you clean them. Then small pieces escape and stick everywhere.

Bob Villa and others recommend bleach or vinegar for controlling mold in a dishwasher. But your infestation looks too far advanced for such mild treatment.

You should use a cleaner called Affresh which was designed by front load washer manufacturers to remove mold and mildew build up.

Unfortunately Affresh and like products are not good at cleaning the mold from porous rubber parts. Try flexing the gasket away from the tub and scrubbing behind it with a stiff toothbrush.

If this doesn't work then you will probably have to replace the door gasket. The part costs around $80.