Why does the shower turn on when I turn on the hot water in the bathroom sink

There is a clog downstream from both of these drains. If they are both at the lowest level of the house and you have a septic system, then it's conceivable that the problem is there, but you'd also see water backing up into that bathroom when you run any other showers/toilets in the house.

Most likely, the clog is in the joint in the drain lines where the shower and toilet join on their way out of the bathroom. Get a snake to fish down the drain (probably best to do this in the shower to avoid damaging the toilet bowl) and expect to go down quite a few feet before you reach the problem.

Edit: Revisiting this question, I'd agree that there's also a likely vent blockage from the bubbling descriptions. But the fact that toilet water backs up into the shower points to a drain blockage, too, and that will be easier to fix.

When you shut off your main (or any valve really but the main is the big one) and turn it back on you can cause debris or sediment that was trapped somewhere in your plumbing to dislodge.

It seems to me that some debris might have gotten trapped in that faucet's aerator. Remove it and see if the pressure seems fine.

If the pressure seems fine with the aerator removed, clean the aerator.

If there is still low pressure try to close and open the shutoff valve for that faucet (with the aerator still removed) to try to dislodge anything that might have gotten stuck in that valve but chances are it's the aerator.