Am I in need of an engine rebuild?

So, here are my issues:

1) If I have the throttle in more than 1/2, after the engine has warmed up, the engine surges.

2) I am loosing coolant somewhere. It is not dripping on the ground.

3) Hard to start when hot. I have to work the throttle to get it going.

4) Once warmed up, will not stay running at idle. I have to work the throttle

5) "Hot" Light comes on. The fan wasn’t working, so I wired a switch.

All of this points to a blown engine. Would I be wrong? Can I get an engine rebuild kit at the dealer?

Not the carb – rebuilt and cleaned and adjusted.

I can do the rebuild.

Its a 1991 Trailboss 350