No need to pour oil in the new filter. The only thing you'll want to do to it is rub oil on the rubber gasket. This will keep it from sticking next time you need to remove it.
As far as running engine after correct amount of oil is in, just run it for a couple of minutes, and you'll be fine.
No need to change oil when engine is warm. If you do it when the engine is warm, you'll likely burn your hands. Let it cool a bit.