This means one person challenges (throws down on to the floor a gauntlet, a type of heavy glove, worn by knights) another to a fight or a joust.
It means to issue a challenge, usually in a very aggressive way, as if you are throwing down a gauntlet.
I believe the idiom you're looking for is "You can't trust a person further than you can throw them". The meaning is meant that you cannot trust someone because you can't throw them very far.
It is when you realize that you can't win and you admit defeat.
"throwing down the gauntlet" means to challenge someone. They used to do it when dueling was popular. Someone would throw down their gauntlet, and if the person took it up, that meant that they accepted the challenge and the fight was on. Today, we don't usually wear gauntlets, but we still challenge each other, and argue... so a "proverbial gauntlet" would be something that someone would use to challenge someone else, or if you said "she threw down the proverbial gauntlet" you would mean that she challenged someone else for leadership or supremacy. "proverbial" means "from a proverb" but sometimes just means old... something from the legends or tales of the past, as it does in this case.
To do work and focus
Distill down, or boil down, as an idiom, means to get to the essence of something, or to simplify it.
To "chuck" means to throw - "chuck it down" would mean to throw something down.
I've never heard that idiom before. Perhaps you mean DOWN AND OUT, which means that the person is at a low point in their life, that they're poor in every way and not likely to make a success at anything in the near future.
I'm not familiar with that as an idiom, so I imagine it means literally throwing jelly at someone.
sit down come over to
The idiom "feeling down in the mouth" means feeling sad or unhappy. It refers to someone looking sorrowful or melancholic, with their mouth turned downwards indicating their emotional state.
This is an idiom meaning to narrow your focus down. It can also mean to narrow your aim and focus on one thing to hit. Picture the zero as a target and you get the idea of the idiom.