State and local building codes will tell you for your area, but, I would use no smaller than 3" to tie in everything.
The toilet is 3 or 4 inch and the shower is 1 1/2 or 2 inch. You could reduce the toilet to 2 inch, but it would never flush correctly or be of much use. The current toilet drain has no trap in that section because one isn't needed. The toilet itself is the trap. If you are using existing waste lines, they should already be connected to the vent stack. If you are adding lines, connecting to the existing stack is fine as long as it is above the last drain.
What is the developed lenght and a VENT STACK should be no less then 4" passing through the roof full size to prevent hoarfrost
in the kitchen cabinet
As long as there is a cleanout on the stack, 25 feet
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.
Which code ? there are several out there
A vent pipe which is the continuation of a SOIL STACK which means a toilet is drained into it versus a waste stack that is continuation of a grey water drain.
A double stack system is where you have the main soil stack to carry away toilet waste (Generally). With a separate pipe taking waste water from a Shower/Bath/Basin to an outside gully leading to a sewage drain (maybe via guttering on your property). Not a common system on new builds but still found on pre 60's plumbing.
Most of the time they are. Depends on how far of course. Usually, the toilet is by the main stack so it only has to go the 12 to 16 inches to get to the drain. The main stack goes through the roof for the vent for the whole house or the bath and laundry. As long as the toilet is getting venting from somewhere, it should still flush.
Yes, BUT check your local code to be sure
The toilet will gurgle as it sucks air through out the toilet trap. You may even hear it suck air through a nearby sink or the tub. Sometimes it may just drain slow or weak.
Shower, cower, bower, power.