For strings, you might try .012, .012, and .021 for melody, middle and bass if you're going to tune CGG or DAA. You might want to go .010, .012, and .021 if you're going to tune CGC or DAD.
For cleaning the wood, if the dulcimer has a lacquer finish it might clean up with some 000 steel wool (rub very lightly). Buff with a soft cloth (old tee-shirt) after rubbing with steel wool. If the scratches are real deep, it may need to be sanded and refinished.
For the bridge, if it will come out of it's slot easily (big IF), you can use a sanding block to sand the bottom of the bone (or whatever material) of the bridge to lower the action. Before sanding, use a yard stick to make sure the fret board is perfectly flat. If it has a high spot, that will require the action to stay high. If it is perfectly flat or has a slight dip in the middle, you can sand the bridge carefully to lower the action. Sand a little, then recheck the action. If possible, you might be able to sand it until a couple of coins just fit under the highest fret without touching the strings (use dimes, pennies, or nickels, depending on your preference for action and as needed to provide clearance for any frets that might be high).
If the tuning pegs will hold the dulcimer in tune, I suggest not doing anything to the pegs. If the pegs won't hold tune, there might be several recipes for applying compounds to the pegs to allow them to keep the instrument in tune. One home-made suggestion is soap and chalk. A local music shop might have some store-bought compounds for friction pegs (violin peg dope).
I hope this helps. Best of luck with this dulcimer.