Well, I can probably put your mind at ease about the freeze up. These engine's cooling systems dont retain any water. Almost all of the water drains out each time the engine is shut off.
I'm not sure what you are describing as far as where you see water dribbling but I suspect it may have something to do with the engine flush out port. Those flush ports are notorious for dribbling while the engine runs. The check valve inside can get a bit cocked to one side and not seal and, sometimes, mineral deposits on the valve and seat will keep it from sealing. You can buy a replacement valve and spring for just a few dollars at boats.net but most of the time you can take the valve out and clean it well enough to get it to seal up.
Popping back through the intake is typically caused by one of three different things.
1. Lean condition
2. Valve timing is off
3. Intake valve tight or hanging open
I agree with racerone that checking and/or adjusting the valves is a good idea. Especially if it hasn't been done before.
Checking the timing belt tension and timing marks is also indicated here. It is a basic, easy task on this engine. Honda timing belts are tough and these little twins rarely jump time but it does happen.
Carbon build up can cause these symptoms so if this engine idles or slow trolls alot then you may consider doing a decarbon procedure.
Many of the 15hp have something called a "fuel chamber" attached to the engine oil dipstick tube. It is a black, rubber cylinder about 1 1/2" diameter and 3" long. The fuel chamber can act as a water separator and fill with condensate. It's good practice to CAREFULLY remove it and shake out all the liquid inside every few months or anytime the engine is hard starting or doesn't run well. Be careful when removing and re-installing the chamber as it is fairly easy to snap the oil dipstick tube off if you use force. A little WD40 for lube helps make it slide on and off much easier.
Just a few ideas before carb removal. Although, yes, these carbs frequently need cleaning when not completely drained and allowed to sit. So, you may need to do that too.
Good luck.