I want to remove a toilet, replace the flooring, and replace with a new toilet. This likely will take a few days. I know to insert a rag in the hole...anything else I need to do? Will having the toilet out of place affect any performance or draining of the washer, bathtub, sinks, or other toilets? Don't want a bunch of water or sludge seeping or gushing out of the empty hole or other toilets not working while this one is out of commission.
If you do use a rag or cloth temporary plug do not forget to remove it before you set the new toilet.
Many times when replacing the flooring in a bathroom it will be a different thickness than the old flooring / sub flooring being replaced. In these instances it can be beneficial to cut back the toilet drain pipe and install a new toilet flange that will come out even with your new floor. In these cases it is common to find new toilet flange fittings that come sealed with a knock out plug that you remove just before setting the new toilet.
If its going to be for a few days and you are living in the house I would recommend a Plug like this: https://www.homedepot.com/p/CHERNE-Econ-O-Grip-4-in-Plug-271543/203308165
They are great and will create a water tight seal. so if there was any issues with a clog further down the line it won't over flow though the plug.
And as Michael said, check your floor height, changing the floor height can cause issues.
On the Wax ring, I like to turn the oven on Warm (less then 200), turn it off, then wait a bit, then put the wax in there for a few to soften it up a little as when you get the new wax rings they can be pretty solid and don't make the best seal.