The air filter appears to have a fuel-oil like film on it, what would be causing this

blocked fuel filter in the fuel line end in the tank or internally corroded fuel tap blocking full fuel flow
the problem indicates insufficient fuel flow causing the mixture to run very lean

This is probably due to not cause prolonged oil changes, oil filter, air filter causing incomplete combustion of fuel, increased engine deposits and other issues.
Recommended oil, oil filters, each with about 5,000 km to be replaced once, air filter and oil filter every 10,000 km driving needs to be replaced, otherwise the air, fuel and oil impurities cause parts wear and clogging the oil, thus affecting the normal operation of the engine.
Use obd2 for automotive diagnostics , you can improve vehicle safety performance and driving more safer.

Exceive fuel is being burned but not so much as to turn the cheak engine light on? How bad is it ? 10 MPG? things to check; Fuel pressure both high or low will cause problems. Timing, most can't be changed , but timing belts can jump off a tooth to standard and it'll have reduced compression less power then burn more fuel. Lack of oil and filter changes, both the oil ( makes the engine drag) ( fume), and the air filter. If you live in a non dusty area you can ty running one tank of feul with the air filter removed , if this helps the filter should be replaced. If it appears dirty replace it anyway.

some models have a fuel pump some are grivitey

FOR oil leak issue and other problems you mentioned check the following help links:---

Radiator leaking? http://repairhelpcenter.blogspot.com/2011/12/radiator-leaking.html

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consuming more fuel? http://repairhelpcenter.blogspot.com/2012/01/toyota-tercel-consuming-more-fuel.html--------- How to maximize fuel economy on your vehicle? http://schematicsdiagram.blogspot.com/2011/11/how-to-maximize-fuel-economy-on-your.html

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Check Engine Light On car? http://schematicsdiagram.blogspot.com/2011/06/check-engine-light-on-car.html

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Common Causes of Car Oil Leaks?

http://schematicsdiagram.blogspot.com/2011/12/common-causes-of-car-oil-leaks.html

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Troubleshoot Car Oil leak?

http://schematicsdiagram.blogspot.com/2011/12/troubleshoot-car-oil-leak.html
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Car oil leak?

http://schematicsdiagram.blogspot.com/2011/12/car-oil-leak.html

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Car oil leaks from valve cover gasket?

http://schematicsdiagram.blogspot.com/2011/12/car-oil-leaks-from-valve-cover-gasket.html

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Rear oil seal leaking?http://schematicsdiagram.blogspot.com/2011/12/rear-oil-seal-leaking.html

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leaking oil from oil pan?
http://howtobyme.blogspot.com/2011/11/kia-sportage-leaking-oil-from-oil-pan.html

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leaking oil from timing cover?http://technoanswers.blogspot.in/2012/02/toyota-corolla-leaking-oil-from-timing.html

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How to Check Differential Oil Leaks and Replace Oil
http://schematicsdiagram.blogspot.com/2011/07/how-to-check-differential-oil-leaks-and.html

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Oil leaks from air filter on ford?http://technoanswers.blogspot.com/2012/01/oil-leaks-from-air-filter-on-ford.html

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---------This will help. Thanks.

cylinders that seize around the exhaust port of the cylinder usually come uner the headind of a lubrication break down ( loss of an oil film between the moving piston and the cylinder bore ) once the oil film is lost, the piston heats up very quickly and expands in the bore, the alluminium is then scraped off the front of the piston, this will gum the piston ring in, the hot gasess from the combustion process then blows past the piston, so the machine has basically seized.
Reasons for a lubrication seizure are.
Insufficent two stroke oil in the fuel.
Poor quaility two stroke oil in the fuel.
Incorrect grade of oil in the fuel.
Lack of maintenance.
Internal bearing break down within the machine.
Over heating due to dirt blocking starter cover air vents, and choked around cylinder finns.
Over heating due to poor chain maintenace ( blunt chain )
Incorrect carb settings ( running too lean )
Blockages in the fuel system causeing over reving.
Working practice ( over loading the saw ).
Air leaks within the machine causeing lean mixture and over reving.
Stale fuel, water in the fuel, dirty fuel.
Hope this answers your question, as you can see there are many reasons for a machine to seize.

Many times when oil appears in your air intake and in the air filter this is usually caused by a plugged pcv valve Positive crankcase ventilation valve it costs anywhere from 3-10 dollars.As far as it not starting my guess is they checked the air filter and left a vacuum hose or airbox open.Some of these vehicle will barely run if an air leak is present at the airbox containing the air filter.Would double check for any vacuum lines that may have been disconnected around the air box and that the box is shut and sealed properly.

Black smoke is burning oil. Blue smoke is usually transmission fluid due to a leaking modulator valve in the older auto transmission. White smoke is due to water getting in to a cylinder from a bad gasket, cracked block or a cracked head. Black smoke from engines on starting and/or running tends to mean there is fuel leaking into the engine, or the fuel mix is incorrect and not atomised correctly. the main causes are normally a nakered air filter that isnt letting enough clean air into the engine to mix. a blocked fuel filter, which is either not letting enough fuel in, or in some cases, a faulty fuel filter which is letting too much fuel into the engine, and causing black smoke due to excess burning. Black smoke is a fuel/air mixture suggestion problems with injection or carbonation. Cause of black smoke is due to running rich in the fuel air mix. Check the oil,the oil leak,the air filter,the fuel pump and fuel filter to confirm the fault.

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it might be what was wrong with my sr125 if you hav been running it on low oil it can cause air bubbles in ur oil pump if so you need to flush ur oil pump and everything will be fine its cause the heat is causing it to seize good luck n i hope this works

Change the fuel filter a gain you may just got dirty or bad fuel. Also check and replace the air filter TDI need a lot of air.