Your first question closely resembles the topic of two pots that are touching (see Rema in SA YD 92:8 and tangentially 93:1). By way of background: two pots are on the fire and touching, one dairy, one meaty. Even when boiling on the fire, their contents remain unaffected. As the Rema writes this is to be avoided a priori. If one of the pot is parve this is permitted a priori if the pots are clean at the point of contact.
However, if there is a liquid between the two pots, many write that the liquid will allow transfer of taste between the pots and render them and the food non-kosher.
Much of the above comes from R Yehuda Spitz's article on the topic, itself a summary of a forty page Hebrew essay he wrote on the topic (viewable here).
R Spitz applies this to your question of placing a hot pot on a meat/dairy surface.
If someone is cooking in a dairy pot and wishes to take it off the
fire, and there is nowhere to place it except on the counter that is
usually reserved for meat items, one may place it there. Nevertheless,
since the issue is not as clear cut as it’s being presented, it is
preferable not to do so, but rather one should place another layer of
separation down first (for example, a board, towel or aluminum foil)
in order to satisfy all opinions.
In your question, the pot is parve which makes it even simpler to permit.
Your second question has too many variations (dry/meat or liquid/milk or frying pan; lechatchila or bdieved) to be answered in this format.
Of course, consult your rabbi
before implementing anything you learn here.