Transducer...
Yes, that is your transducer. It is suppose to be there.
The water you get from the boat when you pull from the water can be normal. During operation, a little water may seep in from your thru hull fitting. Where the shaft passes thru the hull, there is a seal that will leak a bit of water to keep cool. You should also have an auto bilge that keeps it all dry. When you come up on a trailer though, your boat is more inclined than in the water, so you'll have the water that was only 1/4" on the bottom of the boat, collect in back at your plugs....