that dust shield is the clutch/flywheel dust cover. it does what any other dust cover does, reduces foreign debris from entering the bellhousing.

as for the oil on the shift linkage and o2 sensor and all that...it sounds like you have a pretty mean leak. get the car securely on jackstands and have the engine running while you're underneath. if its a major leak, you should be able to see where its coming from. a good way to start is to give everything a good amount of degreaser and some good water pressure. that way you can source the leaks easier.