Re: No cooling in reverse.
Another thing that is unusual with this setup is how high the drive sits in the water. Typically at idle, the water line would be at least at the level of the impeller. Seems like this is setup, the impeller is sitting above the water. This makes it way harder for a raw water pump to keep its prime. Any air leaks at all are going to affect it greatly.
This is fairly important! Most ALL stern drives are designed to sit in the water with a resting water line far
HIGHER (ON THE DRIVE)! It's not as much a problem with a Mercruiser Alpha, (and probably most outboards) because the raw water pump is located physically LOWER in the in lower unit. This puts them in the water so there's water in them when you start the engine.
If you locate the pump higher, like in your case (or with a Bravo, or Volvo) , That makes it MUCH MORE important to ensure that there is NO leaks in the raw water line between the pump and the water line. If there are leaks, then you suck air when you start the engine.............. if it's enough air, THE PUMP DOESN'T prime immediately (or at all) ........if it's just a little air, the pump "runs dry" briefly while water is making it to the pump. This can wear the pump out quickly if it takes too long.
In any case, you can check by replacing the transom raw water supply hose with a clear vinyl hose. Then start the engine and observe looking for bubbles.
If there's no bubbles, then you can disconnect from the t-stat housing and observe flow rate at idle, fast idle etc.....