Won't stay running if you touch throttle and if your lucky and it stays running it won't go up even a little incline.Replaced gas filter,it is getting gas. Replaced also gas throttle cable.

adjust the throttle to max

Remove the play in the throttle by loosening the locking pin on the cable, then pull the cable tight to remove the play and refix the locking nut. Once the play is removed adjust the jet so that its steady at idle.

If you are lucky, you might just need new fuel filter and the fuel bowl cleaned out, and maybe new air filter. If not so lucky you need carburetor rebuilt. Backfire is usaully caused by engine running too lean ( too much air for the amount of fuel present). Fuel bowls get gunk in them and the fuel needles and fuel jets get varnished up. The accelerator pump gets worn out gaskets and/or bellows and doesn't give the right amount of fuel mist when you accelerate.

Full Choke is indicative of a motor starved for fuel

1. Check and/or replace your fuel filter. May have clogged.
2. Make sure the gas switch is on and not on reserve.

GAS ON ? choke on ? clean gas and filter? is it getting gas ? carb out adj? valves out adj? spark plug fouled ?

Could be a clogged fuel filter, or clogged air filter, or trash could've gotten into the carb. Starting with fuel filter 1st and make sure you getting good gas flow to the carb. Does it have a petcock for the gas on/off? Is it operating properly?

clean the carb. look for air leaks.

Remove and clean the carb

pull your plug and see if its wet after it stalls. sounds like its running rich if you're seeing smoke. check the float height. check for clogged air filter.

The sticky stuff is normal. On a lot of dirt bikes and 4 wheelers they have air filters that you are to wash either with a filter cleaner or gas and then you spray them down with a light oil and then wring them out until they have no extra oil in them and the oil helps filter the dirt out of the air heading into the engine. It does look in your picture like it is needing a cleaning.