For most things that are traded they are liquid enough that individual short positions aren't meant to effect the price. They are basically bets based on research that indicates the thing being shorted is overvalued and will fall by itself. Of course people try to help it along by publishing their position and research, but the catalyst is not the short trade itself.

Gamestop went up because people discovered a fairly unique short squeeze opportunity. People piled in enough (through options) which caused a gamma squeeze exacerbating things as market makers needed to buy more stock to hedge their positions.

The 'price' of things is, simply put, the price between the bid and ask order books, so everything being traded has a list of people that will buy at a certain price and people that will sell at a certain price. For example, Stock A's price is 1$, but has people that are willing to buy it at 99c or below and people that are willing to sell at 1.01$ or above. When you buy, you actively fill the order of the guy willing to sell at 1.01$. Say he was the only one willing to sell at that price, the next person willing to sell is at 1.02$, so the spread moves up. If enough people buy, they will fill out the orders and move the price up. Others that were originally willing to sell at those prices may move their orders to a higher price because they see that people are willing to pay more which helps move the price up further. Selling is the same. This way the price is determined by how many people are willing to sell or buy at a certain price, the spread between those prices. Usually things are liquid enough that they are the same price and price movements are gradual.