I believe that you are correct. The "T" looks to be an over flow, and the upper pipe looks to be your water pump in spicket. Is the "T" below or above the water line on the boat. If it is below the water line of the boat you will need to cut the pipe and extend it up above the water line. Otherwise the livewell will just fill up and over flow into the boat.
As for the water pump in spicket, there should have been an aerator pump inline with the old plumbing you had going to the transom. When you turn on the aerator pump it would fill the livewell, and when the water starts to flow into the over flow "T" you'll turn the pump off. (Repeat to add fresh water, as long as the "T" is above the water line the water should circulate.) If it doesn't have an aerator pump I would advise adding one, and get a spray head to attach to the water pump in spicket with a hose. You can mount it horizontally or vertical. This spray head would look similar to this:
The entire system would look similar to this: (Got this from Flow-Rite website - they own this drawing)
As for a pump out, I doubt it had this option. Check to see if you can unscrew the "T" where the nut is. I have a hunch that it is set up this way as a plug. Once the boat is on the trailer you pull the "T" out to drain the livewell. So if this boat is always in water, unfortunately it will always have water in the livewell. Adding a pump out is not out of the question though, just a little more costly. This would allow you to empty the livewell while your boat is still in the water. It would require another aerator pump, a speacial pump out / recirc head from Flow-Rite, and some more plumbing. The cost is really in the pumps, all the flow-rite parts are actually pretty cheap. All in all I think it is worth it, and just ordered my parts to do this to my boat. With the two pumps, plumbing, wiring, thru-hulls, pump out / recirc head, and spray nozzle I I have approx. $150 in it. (I am also adding a automatic aerator timer... that was $55. You could use two toggle switchs and manually control them cheaper. This method should only run you about $100.)
Here is how that system is set up: (Got this from Flow-Rite website - they own this drawing)