What can be done about unusual toilet noise

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.

I'm guessing the valve assembly inside the toilet is finding a resonance when it's just about to close. You can replace it, and it's not too difficult of a DIY job. They sell just the valves:

But if you're going to take it apart to replace the valve, I tend to buy the full kit and replace all the guts.

Note that when my toilets started to leak, it was because of high pressure in the water lines. So it wouldn't hurt to pickup a pressure gauge that screws onto a hose faucet to make sure you don't have a bigger issue. After I fixed one, another would start leaking next as the pressure got a little higher.