gas gets trapped in the drainage system if you don't vent it (and you don't want that gas coming back into the bathroom)
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.
Within 24 "
Yes.
Yes. If not it will be a wet vent and cause MAJOR problems.
partially clogged drain or vent, improperly installed vent or drain
Yes as ALL fixtures need to have the vent placed above the FLOOD LEVEL RIM
Should be individually vented
Yes. You need to open the plumb, & you have an air vent there.
Every toilet should be wet vented with the sink in the bathroom.
As close as possible is best, but there just has to be a vent so that the air that is in the line has somewhere to go so that the liquid isn't trying to compress it. The closer the vent is to the toilet, the better it will flush.
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.
No
Who needs a toilet
Within 24 "
Yes.
Yes. If not it will be a wet vent and cause MAJOR problems.
You can use an AAV for each of these. This prevents having to cut the roof.