I am not at all familiar with a DS80 but the idle speed is normally adjusted down at the carburetor, not up at the twist grip. Normally, you want the twist grip to leave the cable slack when the grip is in the idle position. Otherwise, any flexing of the cable will effect the idle speed, which is not good.
Any bike you bought for $40 is rather suspect from the start. It is highly likely that the bike has a number of problems that are effecting each other. The fact that it doesn't run well with the air filter on tells me that either the air filter is so dirty it can't breath or the system is so delicate that even the slightest restriction causes problems.
Start by cleaning the air filter. It is probably a foam element. Take it off, wash it really good in hot soapy water. Squeeze the water out, set it in the sun to dry.
The carburetor normally attaches to the engine via a rubber boot. If the rubber boot is cracked or has any air leaks you will never get it to run right. Underneath the rubber boot will be the "reed plate". This will be a "tepee" shaped frame that holds flat flexible plates against the frame. The idea is that the reeds will bend one direction, allowing air/fuel to enter but then seal against the frame to prevent it from flowing back. If the reeds are broken or don't lay flat against the frame then you will have nothing but problems.
I would recommend taking the reed plate off and inspecting. If it looks okay put it back, making sure that the reed plate seals reasonably well against the crankcase. Make sure the rubber boot is in good shape and is sealed against the reed plate.
The carburetor probably needs to be cleaned. Unfortunately, most people do not really understand what it takes to clean a carburetor. It is worthless to clean the outside. You can't possibly clean the jets by spraying anything down the throat. The only way to clean what is likely to be clogged is to take the carburetor all the way apart, remove the jets and soak the durable parts in a really nasty solvent (I recommend for overnight). Then rinse in water, blow the passages out with compressed air and reassemble, preferably with new gaskets.
Once you have the clean carburetor on the engine then you need to adjust the idle speed and the idle mixture. Make sure that the twist grip is allowing the throttle slide to move freely to the idle position. Now find and identify the two screws: idle speed and idle mixture. The speed screw will probably have a very blunt tip and extend into the portion of the carburetor where it will block the slide. The mixture screw will have a sharp point. Without any other information available (the manual would be good to have!) I would run the mixture screw all the way in, then back it out three full turns and start there.
Start the bike, warm it up.
Set the idle speed to the lowest speed that the bike will idle, doesn't matter how fast. Now adjust the idle mixture for best idle (idle speed will probably increase). Now back the idle screw out some more to lower the idle speed, again to the lowest speed it will idle. Adjust the mixture screw for best idle. Repeat this process until you get a reasonable idle speed.
Now if it idles okay but "bogs" when you accelerate you may need to adjust the pilot jet or even the main jet. Before doing any of that make sure that you have mixed your gas and oil properly and that you are going to stay with that mix as changing the gas oil mix can change the jetting requirements.
