Ditto to question 1. Don't clean your filters during cycling unless you have to, and then just enough agitation in old tank water to remove grunge, but if you are doing a fishless cycle then there shouldn't be any.
Question 2. Nitrite to nitrate bacteria develops after the ammonia to nitrite bacteria have established enough to provide the 'food' it requires, ie nitrites. The time this stage takes varies but I would keep ammonia up to about 4ppm during this stage to maintain bacteria levels. When nitrites fall and nitrates rise, off the scale normally, then you can water change to reduce the nitrates.
The accepted rule is that 4ppm ammonia is converted to give zero ammonia and zero nitrites and huge nitrates within 24 hrs. Then you are cycled and prior to adding fish do a massive water change, conditioned water of course, to reduce nitrates to below 20.
It is quite common for both types of bacteria to grow simultaneously and for there to be no detectable nitrite reading. This mostly happens if your filter has been seeded with media from another tank. If you have not seeded your filter then it's a bit of a waiting game.
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