Engine backfires when trying to start
 

I have a 1000 that was parked for several months. Now it backfires when I try to start it. It ran ok when I parked it.

Backfires through the carb or the exhaust?

After sitting for awhile fuel starts to dry up inside the carburetor causing a lack of fuel flow (lean condition) and could explain it. Personally I would drain the fuel, put a little bit of fresh stuff in, replace the spark plug and clean/replace the air filter then try again.

When I find myself in your situation I grab my "cocktail" (50/50 mix of Marvel Mystery Oil and Seafoam) and give the fuel a concentrated dose, it works more times than not for me. You can skip the MMO and just use Seafoam with almost the same effect.

If you splash a little gas in the carb to get it started then keep dribbling it in until the motor fully catches itself and is running off the carburetor you should find that it starts running smoother and smoother while the carb is being cleaned out. I find it takes about a half a gallon of fuel with my concentrated cocktail to fully clean the carb.

All of this is assuming it's a dirty carburetor/fuel issue which hopefully it is so it's a cheap and easy fix!

It backfires through the exhaust.

Might be a stuck valve.

Those Engines do occasionally have a situation of stuck exhaust valves as Iron Man mentions.
IIRR removing the spark plug allows one to view the top of the valve.
if so that would allow you to see if the valve is opening/closing while turning the engine over with your hand.
Do remember it is a 4 stroke, so 2 revelations are required to see it open/close.

engine backfires when trying to start
 

I removed the spark plug, and it looks like the exhaust valve is not moving when I turn the engine over by hand.
Is there a way to unstick it without removing the head?

Take off valve cover on side of block, attempt to free. :beerchug:

Keep in mind the Kohler motor has an ACR (Automatic Compression Release). That means the exhaust valve will be slightly open to help ease starting. Once engine is running, the exhaust valve closes fully. The big point is, don't go smacking down the exhaust valve with a hammer in attempt to close the valve fully. You can break the tangs if you're not careful.

Spray some of yer favorite penetratingly elixir on the valve head through the spark plug hole multiple times during the day, then come back the next day and see if it will pop loose.
as mentioned, don't whack on it with a hammer because the compression release damage if so equipped with one.( I don't remember if a 10 hp has one.)
The valve adjustment compartment is a better place to tap/pry with a screwdriver.
If you understand the operation of the engine,
about 1/2 way down of the piston after the plug fires, should be safe as the compression release would not be holding the valve open, as it will be on the power stroke.
A little tap tap on the valve head through the plug hole will not hurt it :bigthink:

engine backfires when trying to start
 

I took of the side plate and I can see both valves going up and down, but when I look through the spark plug hole nothing seems to be moving. I placed a dowel rod on the valve head through the spark plug hole and I could not feel it go up and down.