Turn off the water, flush the toilet (keeping the handle down to empty the tank), then start plunging the bowl with your plunger to pump most of the water out of the bowl. Now disconnect the water hose from underneath the tank, take off the tank lid, crouch over the toilet, facing it, reach down and grab it on the flat portion of the bowl that's between the tank and the bowl (so you're actually grabbing the bowl--this point is the middle/balance point of the toilet) and lift up. You'll have to do a bit of a "crab walk" with it to move it (so make sure no one's watching you...). If you have a shower in that bathroom, carefully lay the toilet in the shower (maybe put some towels down first). If no shower, lay some towels or newspapers on the floor...there won't be that much water splashing out when you lay the toilet on it's side. Now put on your latex gloves, get a flashlight (and a small mirror if you have one) and start looking in the top AND the bottom of the bowl for the obstruction. If it's stuck in the middle of the toilet you may or may not be able to snag it with a wire coat hanger. If not, you're going to have to buy a new bowl (they're not too expensive) and change out the old bowl with the new, because short of breaking out the porcelain, sometimes there's just no way of getting things out of a toilet bowl.
When replacing the toilet on the flange, check to see if the bolts have a nut securing them to the metal flange. If not, put the NEW bolts from the wax ring package on them to secure them (due to the wet environment under the toilet, the new bolts will last longer than the possibly rusty ones that were holding the toilet down). That way, when you put the toilet back over the bolts they won't move around and when you go to tighten down the bolts securing the toilet there's no possibility that the bolts will come loose and you'll have to pull off the toilet and start all over again. Reconnect the hose (hand-tight is usually enough), turn on the water and feel around the bottom of the tank for leaks as it fills with water. BTW, rub the black washer inside the end of the hose with your finger when it's off. If the rubber rubs off on your finger, it's time to replace the hose ($3-$4).
Unless a customer specifically requests it, as a rule I don't caulk around the base of the toilet for the simple reason that if the ring compresses enough to let water leak underneath the toilet, I'd much rather see it on the floor surrounding the toilet almost immediately than have it sit there, soaking through the floor and causing all sorts of floor damage for who knows how long before the leak is discovered (can you say "wood rot" and "mold damage"?).
Hope this helps.