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.