2003 pt cruiser, seems like oil is coming out of one of the spark plug wires where the engine is

Check spark plugs and plug wires.
Have you changed the fuel filter lately? Dirty air filter will make it run rough. These are possibilitys only.

Fuel is metered based on a signal from the coolant temperature sensor. When the sensor fails the engine is slow to start.When the sensor gives a false signal of over heating it causes problems like yours

If plugs helped...the problem went away...then it could be a plug getting fouled by excess fuel or oil. Look at your new plugs and see if they are discolored or went with gas. could be an injector problem. It could also be an ignition problem. You moved the plug wires around and they worked for a few minutes till the new "connection in the wire" burned up. Check your plug wires with an ohm meter. Check your plugs for fuel or oil on plug...a sign of stuck injector, oil leak, misfire. Did it really help or was it just wishful thinking?

Does the vehicle turn off, run rough, or hesitate?

It sounds as though you have a faulty Idle Air Control Valve".

Crank sensor controls fuel not sprark. Check the cam sensor, this sensor is usually on the head of the engine. Check for this sensor on the top left or right sides of the engine (it depends if turbo/none turbo). Another issue that may cause no spark is failed ignition coil. The ignition coil is on top of the engine, with 4 wires routing to the spark plugs. Hopefully this helps!!

pcv cracked? The spark plugs are installed into the top of the head.
Replace cracked or dry rotted hose or the gromit for the crank case pressure valve, this should return through a line to the intake. Good luck

Use the correct champion plugs for starters, make sure the wire is firing. If the wire is firing, the coil pack is good. You will need to check compression and fuel delivery after that.

The P0302 Code points to a misfire detected on the # 2 cylinder. If its stumbling, I would first suspect the plug or wire is bad. I believe the ignition coil on that vehicle is a single coil pack with four terminals, so if only the P0302 code is coming up I would start with the plug and wire. May be time to do a tune up and just replace all the splugs and spark plug wires.

Spray cold engine down with carb cleaner or brake cleaner. Let air out. Keep watch on where you first see the oil. It is most likely from your valve cover gasket.

I know that the PT's, Neon's and Stratus's shake when in gear at a stop. It maybe possible that its time for a tune up aka Spark plugs and Plug wires.