FWIW, if you're worried about adding more water to your floor, just use a sponge and a bucket to get the water out. If you're going to pull the toilet, you might as well, though, turn off the supply, then hold the flush lever down until most of the water leaves the toilet. Rubber gloves are nice, but a toilet bowl isn't as dirty as many people think. Not saying I'd want to drink it, but if you don't have any cuts or scrapes and you wash your hands later, it's not a big deal.
Try to remove any sealant around the base of the toilet first, then it might be easier to tell where it's leaking from.
Note if the toilet can be rocked back and forth at all. If it can, you MUST shim it prior to installing a new wax ring and setting the toilet. You shouldn't rely on the bolts to prevent rocking, as that can tend to lead to breaking your toilet, or leaks. I usually just use some coins rather than buying special shims, but whatever works for you. For me, it saves both money and time over a trip to the store.
One thing that can blow out a toilet is if you were really aggressive with plunging and you did it incorrectly. Many people think the main action is to push down hard. That just tends to jam things in tighter. The better way is to slowly push down, then rapidly pull up. That helps to rearrange things so that they can then flow down. If the toilet flange was not installed on top of the finished flooring, whomever installed the toilet may not have used enough wax to fill the space, or the wax shifted while the toilet was being set down. It's best to place the wax on the flange, then set the toilet onto it. A lot of the instructions say to put the wax on the toilet, then set it down, but the wax can shift much more easily that way. There's a 'sticky' on how to install a toilet above...read that and you shouldn't have any issues.
You do not need to remove the tank, but the end result is you'll have to lift more. If that isn't an issue, leave it attached. Otherwise, you may need to remove it so you can lift it. IF you do, you might want to get some new rubber washers.