Depends on local codes and developed length
yes
The drains for the shower, toilet and sink all connect to a common line. Assuming the toilet is not backing up, the problem indicates that there may be a partial blockage, enough so that some of the toilet flow is backing up through the shower drain line. You need plumbing repair. The shower leak you mention should only be in the shower, or else you have a broken drain line as well.
Depending what you are wanting to supply. Basin, shower, toilet, bidet, kitchen can be fed with a half inch line Main line and bath can be fed with a three quarter inch line
A down line clog would cause this. The shower drain is lower than any other drain in the bathroom so it is the first to tell you of a clog in the line. Time to Roto Rooter! Y-THINK-Y The toilet and shower must have the same drain. There is a blockage somewhere after the toilet and shower drains meet. The backup takes the easiest route back which must be the shower drain. Try snaking out both drains. Start with the toilet. To do the best job, remove the toilet and start from floor level.
Run a new cold supply line from the source. This will drain less pressure from the the cold shower line.
Your drain line is plugged somewhere after your shower and toilet line are connected. The reason for the gurgle in your toilet is because air is trapped in the drain and your toilet is the closest place for air to flow to. If you had a vent on your toilet trap arm then it would gurgle in the vent rather in your toilet but your drain is still plugged and needs to be snaked out and maybe your drain line needs to be examined for right grade or broken.
usally a line stoppage
Snake the downstream line and get rid of the plug up.
You cut the drain line and add a 2" section.
Not many. Maybe one at best. Half inch copper line is too small for most needs. 3/4 only costs a little more. You will go completely mad waiting for a bathtub to fill with half inch lines. Every flush of a toilet will make the shower die out.
Because the two pipes are connected, and the shower is farther down the line than the toilet. So when you flush, you are hearing the water from the toilet pass by your tub. This is also why you will have sewage in your tub if your waste water line is extremely backed up.
In my experience I've seen this type of scenario in older buildings. Usually the sink will tie into the stack in the wall while the shower generaly connects in below the toilet also known as the closet bend. In alot of older houses and apartment complexs they have tied multiple drains together in a smaller pipe before going to a larger one. For example the lav with the tub, the kitchen sink with a tub, or all three together.
It might mean that there is air in the line. Maybe install a water hammer.