How does mining set a floor?

let's assume it costs $4k to mine a bitcoin. If the price were to fall to $10 or some extreme low value, the miners would see no reason to mine. Therefore all activity would stop. But in order for existing bitcoin holders and merchants to use bitcoin, they need transactions confirmed. So this will lead to the price rising to an equilibrium where the miners will mine at a loss and merchants and hodlers will sell at a loss which is optimal for both than if the price is too low.

cost to hash, in electricity. If the cost of mining is greater than the price of the coin, the miners turn off their hashboards. Some like to think of this as a floor, however if everyone turned off their miners the system would fail to produce blocks. The network difficulty will take 2 weeks to adjust, and the miners will start mining again if it is profitable.

If the difficulty adjusts down to make mining profitable, that's because the miner is earning more coins, but each coin is still worth less than before.

Not necessarily. While there is a positive correlation to hashrate and price, the market sets the value of the coin.

Just because miners are making more or less coins does not make them worth less or more. Miners choose when and if they release the coins to the market.

It doesn't, it's the other way around. Price sets a floor on hash rate.