Heck that dosent look too bad compared to some I've seen you can probably get away with all the parts that are on there.
Go ahead and check the airfilter out unscrew the thumbscrew on the black plastic thing and pull the cover off once the screw is removed. Make sure to remove any debris inside before removing the air filter which will be a piece of foam and then take out the air filter and make sure it is clean( you may need to replace that).
I would definitely check out that muffler something about it doesn't look right to me like it is missing something. To be on the safeside I would remove the muffler and check it out to see if there is damage and clean the carbon.
Make sure to check the oil and to er on the side of caution change the oil you don't know where its been.
Also like others have said open the fuel tank and shine a light (not fire) down there and check if it is dirty or rusty in there if it looks good you are good to go. Make sure to remove the old gas( you can disconnect the fuel line to drain) and fill it up.
A 4 cycle like the one you have can be started without being mounted to a piece of equipment just put it down on something soft like grass and hopefully your janky rigged pull start will accept a piece of dowel so you can start it up.
I've never used a small gas engine before, so I don't know what that sounds like.
A motor like that as you are pulling the cord should feel easy at most parts of the draw stroke and then the pull will become good and stiff at the compression stroke of the motor.
It should sound like it is sucking and blowing real good I'm sorry I don't know how else to describe it.
edit:Upon closer inspection of your pullstart it looks like it is missing a screw go ahead and make sure that the pull start is securely on the rest of the motor for safety the oil on the ground reminds me of a honda I took in with a cracked crankcase hopefully you just played with the oil caps.