Go ahead and remove the toilet. Its a nice job to break the wife in on. Turn off the water and dip out any excess. Have some newspapers to set it on since the old was ring will be stuck to the bottom. Also be prepared for a little water to spill since it all don't dip out. Then run a snake through the line. Have the new bolts and wax ring ready.
The bubbles tells me that there may be a plug around the vent pipe for the toilet unless it wasn't vented right to start with. Could of had a squirrel, mouse/rat, or bird end up dead in the vent and plugged it. Also leaves and small limbs do a good job also. Plumbers sometimes start their snake from the vents on the roof if they have to. They usually try to run the garden hose down the vent on the roof to see if it washes the obstruction down first. Try this before pulling the toilet.
Knew a little kid once(won't name names here) who used the vent pipe for shooting baskets with rocks. When the toilet wouldn't flush right, his mother called his uncle the plumber. After crawling through a muddy crawlspace to fix it, his uncle was extremely unhappy to find a bunch or rocks when he opened the cleanout. Its a good thing he never found out who did it or that kid would have gotten a paddling.