Yes, sometimes I open the packet with a knife and use that to push but thinking on it that introduces bacteria from the packaging.

I use paper towel as a utensil rest or hang the working end off the edge but might stand it vertical in a Pyrex jug

Nothing goes between raw and cooked. I’ve got a 3 set of dedicated chopping boards I don’t mix. I hate my fridge because the coldest compartment for fish/dairy/meat is at the very top and the vegetable crisper at the bottom! You’re not meant to have raw meat stored over other foods 😬 (Seriously why is raw meat stored over the salad)