Hi. There are a couple ways coolant can get to the bellhousing. The rear bolts on the intake can corrode and break off. This releases presure on the gasket and the coolant leaks out around the rear water jacket. Check the rear corners of the intake from underneath. There are freeze plugs along the side of the black. When these plugs leak the coolant will run down, onto the rail of the oil pan, run along the pan and dump on the bellhousing. Check forward of the bellshousing and see if you can see a trail. There are also two freeze plugs behind the flexplate on the back of the engine. These are normally one of the first ones to go because they are right at the bottom of the water jacket and corrosion starts on them first because it builds up from the bottom of the water jacket.
The running problem may be a few things. Make sure the engine is in good tune. Plugs, wires, cap, rotor should all be checked or replaced. If you have to keep the throttle open until it warms up to keep it running it normally indicates a bad idle air control motor. If it runs ok for a couple minutes and then starts to run bad until it gets hotter it may be a bad upstream oxygen sensor.