My group uses very contest check against monsters, NPCs and other characters. I would like to know the skill or attribute, must be used if the character is wanting ...
a) Intimidation vs ......?
b) ....... vs Insight?
c) Perception ....... vs?
d) Deception vs .......?
And in case you erase your footprints and someone want to trace them ... would be survival or nature vs ...?
The relevant rules are:
How to Play, Basic Rules, page 3
Ability Checks, Basic Rules, page 58
Contests, Basic Rules, page 58
My ruling is:
It depends on the player's stated goal and approach and the challenge represented by the NPCs, monsters, or obstacles. According to the basic conversation of the game, players may only describe what they want to do in the face of the DM describing the environment. They cannot choose to "make a skill check." Only the DM may decide when a skill check is required and that's when he or she feels the outcome of the player's described action is uncertain such that a result of the adventurer's cannot be narrated until the die is cast. If he or does find it is uncertain, the DM sets a fair DC or calls for a contest if the character's efforts are being directly opposed by the NPC or monster.
So, for example, let's say one of your players has established fiction that amounts to trying to force a group of NPCs into backing down from a fight by using overt threats and demonstrating martial prowess. You have decided that because of the circumstances, the result of this action is uncertain. Now you must decide to assign a DC or call for a contest. But what are these NPCs doing to actively resist the intimidating actions of the character? You could say that one of the NPCs in particular is responding to the character's threats without fear, trying to incite his comrades to violence with a commanding personality. Here you might then make it a contest of the PC's Charisma (Deception) check and the NPC's Charisma check or even Charisma (Persuasion) check, depending on the approach you describe. Short of this though, you might just make it easy and set a DC, straight-up. It really depends on what is going on in the scene - and the same goes for the rest of the ability checks and skills you mentioned above.
Just remember, not every action calls for an ability check, only the uncertain ones do. If you say that every action is uncertain, you run the risk of diminishing roleplaying by making the players' decisions matter less and their rolls matter more. A careful balance must be maintained.
But you or your DM may make a different ruling based on this or other criteria.