Loose or Sticking Toilet Handle
Lift the toilet lid and adjust the lift wires. Also, you may want to detach, clean, and reattach the handle. Handles can get pretty gunked up over the years, which leads to sticking.Often, it's easier simply to buy a new handle and install that.
If the problem is that the toilet doesn't flush completely, you may need to adjust the water level in the tank.
Toilet Running Continuously
Several things might cause this. First thing to do is lift the lid and examine the lift wires or chain. An adjustment of the wire/chain mechanism is usually enough to fix the problem.The toilet float ball sometimes can become cracked over the years, allowing water to seep in. If the ball float is not floating correctly, it will allow the toilet to run continuously. Pick up an inexpensive float ball at the hardware store and replace.
The wire that attaches the float ball to the ballcock can easily be re-bent. Try that.
Finally, the ballcock - the operative part to which the float ball is attached - may need adjustment, repair, or replacement. Experiment first by adjusting the ballcock before replacing.
Water on Floor Around Toilet
Not as disasterous as it sounds. Usually this is a simple fix.First, examine the tank to see if there is any condensation or "sweating." If so - and if the condensation is dripping on the floor - you've found it. Insulation kits are available which install within the tank.
Alternatively, the tank may be cracked, also allowing water to slide invisibly down the porcelain to the floor.
The bolts which hold the toilet to the floor are easy to tighten if you believe a loose toilet is the cause (don't tighten too hard, or you can crack the toilet).
Finally, if need be, you can always remove the toilet and replace the wax ring that acts as a gasket between the toilet and floor. This wax typically gets dried and hard over the years, allowing water to escape.

