A foam air filter is designed to have a little bit of oil in it to catch the dust but if you were actually able to squeeze out a large amout of oil and it looks like black engine oil the problem may be a faulty breather thus allowing oil to flow through the breather tube and clog the filter. Have you flipped the mower over recently? If so that would cause oil to flow into the filter.
As far as your carburetor goes if you have a plastic tank mounted carb and you have the fair filter removed you should definitally be able to see gas being shot into the carb when you push the primer. If you can't see the fuel try tightening the screws the hold the carb on the gas tank. If this doesn't work take off the carb and clean it. Make sure when you put the carb on you have the gaskets in the same order that you took them off.
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Push Lawn Mowers | The Practical Consumer