1. Based on the description of what a player is doing, both a tool proficiency and a skill proficiency are integral in what they describe they are doing, and both applicable. Do you pick one, have them clarify whihc is primary, add both/grant advantage to one, do something else? Especially when the bonuses are not the same like only one proficient, or cases like expertise/magic tools/etc.
In general, if more than one skill applies, they can choose which to use - in the vast majority of cases, that means they'll use the best applicable skill. See (3) below for a reason they may not want to do so.
2. One character examines something, gets a poor roll, calls over another who wants to roll, and then they call over more if the rolls continue to be bad. Since they are being done one at a time it's not the case of one character helping another. So effectively everyone gets a separate roll which is almost like penta-advantage for a 5 person party.
If they have time for it, sure, why not?
3. A character makes truthful statements in ways that are deceiving. Like "you can have all the gold I'm carrying if you let them go" when they have little on them. They see it as persuasion because the character is truthful they will gladly give you all they have, but since the player intent is to get off cheap, it's like they are attempting to deceive to imply they have more.
First, there's the question of the approach (if stated by the player)
"Please, sir, I don't have much, but what I do have is yours if you let them go," is Persuasion.
"Dude, listen, I have a whole pouch of coins here. You can have it
all if you let them go," when the pouch is full of coppers and iron blanks, is Deception.
If the player isn't the sort to outright demonstrate their approach, then this is like (1) above - more than one skill may apply. And, here's the difference that may drive a player to choose one over the other.
If you use Deception: you are perhaps more likely to succeed (because you actively try to make it seem like a lot of money) but the target will swiftly learn you were being dishonest, and is apt to not react well at all.
If you use Persuasion: you are perhaps less likely to succeed, because you don't actively misrepresent how much you have, but the target will not feel cheated when the deal is done.