1. Starts easily when cold, but after the start it won't be able to start again until cold.

2. Water at the rear is not shooting right up as it suppose to be.

How can I fix this problems?

First find out compression readings. Most likely the engine is worn out and doesn't have enough compression once it warms up to stay running. As the metals expand and open up the tollerances it loses compression. you might need a top end rebuild. Where are you located at in the country? How many hours on the ski/Engine?