Depending on cage size you would need to clean the cage fully every month or less.For example,80 cm x 50 cm cages could go around 1-2 months before a full clean if they have deep substrate and a toilet is used.You could probably get away with the same schedule for spot cleaning since you know that works,though.
For the toilet,weekly washing with warm water should be fine and then every few cleanings of that you can add the washing up liquid.Same goes for the jar!As long as you scoop out droppings it's fine.For the wheel,you really don't need to do much more than wipe it dry every few days if necessary and then every two weeks you can wash it.The paper is a good precaution as accidents do happen even if not urine related.
Disinfectant sprays only really need to be used when an animal is ill.Soap and water or just water is perfectly fine for healthy animals,and you only really need full cleans every 3-6 months if you have a good size cage.Partial cleans monthly could work as well,and would help to keep stress low.
Hamsters do hoard food as well as droppings,but don't worry about the droppings.It seems gross,but some small animals actually get nutrients from a different type of dropping that is actually very beneficial to them.Do not remove them,as they are perfectly fine and even beneficial.Keep the hoard as well,only remove something if it looks close to spoiling.
I actually like the length of the post!You're very thorough and that's always good
Never be afraid to have big posts with tons of questions,since you get to learn and it can also help other hamster owners as well.