Who needs a toilet
Yes.
No, that is why it is called a vent pipe not a soil line
Yes if the vent pipe is 4" and vertical by code and there is nothing connected between your toilet tee and where you would connect your new toilet tee. If not worried about codes then yes you can connect to a 3" vent line if no other vent lines are connected down stream. If you have a vent pipe that is less then 3" and or you have other vents connected between your old toilet tee and your new toilet tee then NO you can not connect a new tee for your toilet.
1-Clear the obstruction from the vent or drain pipe 2-Be sure the vent is properly sized and graded up towards the penetration of the roof, or where it ties into the vent stack.
A toilet has to be vented otherwise you are trying to force liquid against the air in the drain and it is pulling against a vacuum. It will never flush right.
Any vent is usually the size of the drain or waste pipe it is connected to. 3 or 4 inch on a toilet.
As waste falls down a pipe it creates a positive pressure in front of it & a negative pressure behind it. To allow waste to flow down a pipe properly these presures are balanced via a soil vent pipe. Im not a builder, but sounds like there may not be a soil vent pipe on your system, or at least an 'air admittance valve' (AAV can sometimes be used if the soil pipe cant be taken externally).
No, it has to be vented
No I just checked and mine has no condensation, thanks fro asking
The highest 4 inch pipe should have a vent on it, or directly above a toilet junction Above the flood level rim of the fixture it is servicing IT does not mean the toilet
Your vent pipe for that part of the plumbing is either blocked or non-existent. Having to tip the toilet off center slightly is letting air enter into the system, something that should be happening under normal conditions if the vent pipe were working properly. That vent pipe needs to be snaked out and cleaned.
Normally 11/2" DEPENDING on developed length of piping and fittings