The wax ring underneath it? Shut the water off, flush the toilet and let as much of the water go out as possible. Get the remainder out with a sponge or towel. Dip the water out of the bowl. Take the two nuts loose on the base of the toilet on either side at the bottom. Disconnect the water supply line also. You can just tip the toilet forward to get the old ring out and to put the new one in. There is still water in the bottom of the toilet so don't move it around a lot. Put the wax ring either on the floor around the drain or on the toilet. Wax side against the toilet and the black funnel into the drain. Lower the toilet back down making sure the bolts come through the holes. Stand over the toilet and put both hands on either side right behind the seat and put your weight on it. This will flatten the wax out and seal the toilet down. Now put the nuts back on the bolts and tighten them. Gently a little on each side until they are snug but do not over tighten them because the base will break.
If you reduce the soil connection to 2" and install a trap
You need at least a 3 in. pipe -- shower is 2 in. pipe
if you mean the toilet floor gasket, that is about 1 hour labour. Call 3 local plumbers and ask for rate
You can install the vanity first. Then, install the tile up to the vanity and around the pipe for the toilet. Then, install the toilet. That way, you do not have to replace the tile should you ever have to replace the toilet.
Yes
Yes, you would indeed need a permit to install another toilet. They are easy to get.
The time it takes to assemble and install a new toilet is about one hour.
No, toilet seats are not hard to install. There are two bolts that go through the rear of the opening on the toilet and when these bolts are secured, the seat is installed.
If you do not want to change the drain pipes, you will need to install a toilet that matches the existing rough-in. If you can find a brand name and model name or number you can order an exact replacement or cross-reference to find a fit.
The same as you would any other. Most often used when replacing an older toilet that is 14 inches out from the wall with the modern ones that are 12 inches out. How you attach it to the existing drain will depend on the type of pipe you have and your particular circumstances.
The name for the wax seal for a toilet is a wax toilet ring. Some people also call this item a toilet wax bowl gasket.
When you purchase a new toilet seat, it is going to come with the instruction manual as to how to install the seat, when you get it in your home for installation.