I could be wrong, but is it possible you have a California bike that had it's evap system removed? That rubber cap sounds like it goes to the evap canister fuel return tap, which is on the right rear bottom of the tank. If this is the case, then you need to leave that rubber cap in place and see if you also have a drain hose that connects to the right side water drain hole under the gas cap and an additional breather hose. The California bike would have an additional breather hole under the gas cap on the left side. The tank peeling sounds like a bad cap gasket, which you can replace.

 

The ZR-7 probably is not a Calif model. Under the tank toward the front is the fuel gauge housing. It has a port for a hose. Currently, there is not a hose connected to this port. Once I attach a hose -- where does the hose go?
I will re-cap the port at the rear of the tank. Also, I've ordered a new fuel cap assembly.

In addition, on the far right carb at the top is a port for a hose that goes to the Air Suction Valve. The port has been closed off with a short hose and bolt plugging the hose.
Where is the Air Suction Valve located? (it appears to be in front of the carbs?) Should I try and attach a hose to this carb once I locate the Air Suction Valve?

 

Okay the fuel gauge housing is more toward the middle/rear of the tank on the right side. It's square and black. It has a breather hose that runs under the tank along the frame and goes to the ground behind the right heel plate.

The air suction valve is under the tank near the front, and is attached to the head covers via a left and right hose. There is a port for a hose pointing down. I think you are confusing this for the fuel gauge housing. This is the air suction valve and that port connects to the port on the right carb. That solves two of your connections.

I'll go double check that port that leaked on you. It's COLD here!

UPDATE: That's the air vent for the tank, which is supposed to have a hose that follows the overflow hose to ground. So if you're overflowing fuel from that while running the engine, then the tank probably isn't venting properly. It's possible that your new gas cap and gasket assembly will solve that. Otherwise, someone else might have a better explanation for you on that one.

 

Thank you. I’m hoping the new fuel cap will solve the fuel seeping problem. I think I have identified the fuel gauge housing. In addition to the port for the hose it also has an electrical connection extending from two wires exiting the housing. This wire connects to another wire. I’m sure I can locate the air suction valve. I’ll unplug the hose on the right carb and connect it to the air suction valve (as soon as it gets warmer).
It’s cold here too!!