yes, the melted plastic piece in your 2nd pic is #9 in the diagram called the exhaust valve.
http://partsfinder.onlinemicrofiche..._oem/Seadoo_PWC.asp?Type=18&A=60&B=5&Action=O
is that ring on the right of your first pic a rubber o-ring or a spring ring? Stock, there is nothing around the base of the o-ring (note nothing in the parts diagram) some people put rubber o-rings or small zip ties. I don't put anything around the base on mine and never had a problem.
the valve on the rear of the engine you can fully assemble in your hand then stick in in one piece (excluding the spring and the black cap with the red button).
the valve on the front you have to put the gasket on the rave and put it in the slot, then attach the bellows to the base and put the bolts in. Then you screw the plastic exhaust valve onto the top of the rave. the bellows will usually seat itself in the groove on the exhaust valve once you get it screwed tight. you may have to help it a bit. the hardest part is then getting the spring ring around the top of the bellows, that takes some practice.
The o-ring I was asking about in the prior post is #4 in the diagram. I'm betting that is what caused your valve to melt. when i got my skis, one of the valves was stuck and melted like yours and there were just remnants of the o-ring remaining.
can't tell from the picture if the inside of your bellows is melted and needs replaced or if that is just oil