Yes.
>Each miner will be competing for the miner's fee, and will buy more electricity if they can mine more blocks (and get more miner's fees).
Yes.
>The rewards will be proportional to the market cap of bitcoin. If it goes up, so will the investment in electricity.
No.
You're conflating two different things here: A reward-dominated bitcoin and a fee-dominated bitcoin.
The reward takes money from bitcoin owners in the form of inflation. It decreases the market cap by increasing supply.
The fees take money directly from bitcoin users.
In the reward-dominated case, miners are funded by new demand for bitcoin, which props the market cap up. In the Fee-dominated case, miners are funded by bitcoin users who compete with each other for limited bandwidth. In either case, the money being given to miners by these two parties is always greater than or equal to the money spent on mining (power, hardware).
What we've seen so far is a reward-dominated bitcoin in which demand is very high and the market cap increases as the block reward decreases. I predict the demand will stabilize and the reward will continue to decrease. This means there will be less mining. Eventually the reward will asymptotically trend to zero in which case bitcoin will become fee-dominated.
Fees are dependent on the supply/demand for bitcoin bandwidth (the supply is a static 1MB/10 mins), not market cap or price.