Just moved Saturday (12.24.11), everything was good. Came home (12.27.11) and toilet was gurgling/bubbling (assuming from air) It overflowed and then went down on its own (OVER & OVER) Some water came up through tub. It's public sanitation (not a septic tank) City water. Sump pump in basement would turn on each time that toilet would do this, and water was pouring from a pipe that runs in between the wall and the sump pump. Help!

Sounds like you have a blockage downstream from the tub that needs to be snaked out. This can best be done from the tub overflow,(see image).
Unscrew and remove the overflow plate,(The one with the drain lever) and pull out the tub stopper. You now have a clear shot at the trap. The way you were going in through the drain it ran you square into a tee that you can't get around. Now feed the cable( 1/4" best or 3/3" with a small tip. You can rent a Ridgid K50 sink cable.) About 2' down you will run into the bend of the trap. Crank and push at the same time to work around it. Your blockage will be found in the lateral pipe going to the stack. You shouldn't have to put out more then 10 or 12 feet. I have found tub and shower clogs to be mostly hair and the worm should auger into it and pull it back. After you snake pour a few pans of boiling water down the drain. This will loosen the grease and flush the mess out into the main. Hope this helps and thank you for rating my reply. TOM