You're going to have to inspect each junction and device in the circuit, starting with the first know bad device. It sounds like you have a GFCI in the house, which feeds a receptacle in the shed, which in turn feeds a switched light in the shed. Since the receptacle in the shed is the first know bad device, you'll want to start there.
Turn off the power at the circuit breaker. Pull the receptacle out of the box, leaving the wires connected. Inspect the wiring for loose connections, and fix any found. If there where no loose connections, you'll want to determine which wires entering the box feed the receptacle. If this is not obvious, you'll have to do some testing. For this you'll need a multimeter with a continuity function. Disconnect one of the neutral (White) wires from the receptacle, set the multimeter to test continuity, touch one probe to the equipment ground, touch the other to the disconnected neutral, then to the still connected neutral. If one tests positive (audible tone, or reading on display), that is the feed line. If neither or both test positive, you'll have to do testing with the power on (which I won't cover here since it is unsafe, and you should call an Electrician at this point).
Now that you've found your line wires, reconnect the neutral to the receptacle. Leave the receptacle out of the box; making sure nothing is in contact with the edge of the box, then turn the power back on. Turn the light on in the shed. If it goes out, use a non-contact voltage tester to test for voltage. If you have voltage on the line but not the load, replace the receptacle. If you have voltage at neither point, run similar tests on the GFCI receptacle.
If you have voltage on the load side of the GFCI, but no voltage on the line side of the receptacle in the shed. You'll have to inspect the wires connecting the two devices. Start by shutting off the power, at the breaker. Next disconnect the load wires from the GFCI, and the line wires at the receptacle. Run a continuity test between all combinations of the wires, none should test positive. Finally connect one pair of wires at one end (so in the shed, twist the black and white wires together), do a continuity test on the other end to make sure the circuit is not open.
If all of this has been done, and the problem persists, you'll want to contact an Electrician.
As always, if you don't have the proper tools, or you don't feel comfortable doing any of the work mentioned. Do not hesitate to contact a local licensed Electrician.
Yes, that is the entire purpose of the load terminals of the GFCI. You can imagine that the protection element of the GFCI extends both to its receptacles, and then just imagine the load pair as another "receptacle".
As to the second part, yes, if you just connect the terminals together "before" the GFCI protection is in place, the GFCI will have no effect - remember, it can only sense problems on its own receptacles and its load terminals.
Best Answer
Yes, you can do that, using the LOAD side of the GFCI.
But not only can you do it, it is actually a very good idea. That is because GFCI includes sensitive electronics not in a regular receptacle. Keeping a GFCI clean and dry and temperature controlled will likely make it last much longer than it would outside.