Use your non contact tester, and start tracing the circuit back to the panel. Check each device along the way, and keep a close eye out for GFCI receptacles that have been tripped.
You may want to pick up a multimeter, and test for voltage at the switch, and at the fixture. Water could have entered the fixture housing, and fried the motion sensor. So make sure you check the wires feeding the fixture.
If you don't feel comfortable working on live lines, call in a licensed electrician to have a look.
It's difficult to answer this question, without actually seeing the plumbing leading to each fixture. Reductions in pipe size, differences in pipe material, pipe length, number of elbows/tees, etc. These could all contribute to a reduced flow.
If you had a leak large enough to reduce the flow by half, you'd likely either notice it as water damage, or on your water bill. Reduced flow being the only symptom, a leak would be low on my list of things to check.
I'd likely start with the sillcock itself, to make sure it's operating properly. If the valve isn't opening all the way, or it's gummed up with junk, you'd definitely see a reduction in flow. Turn off the water to the fixture, and remove the sillcock.
NOTE: That's a frost-proof anti-siphon sillcock, so it actually connects to the supply plumbing somewhere inside the house. So you'll want to have a bucket under that joint, to catch any drippings when you remove the sillcock.
If everything checks out there, start working your way back along the supply. Look for pipe size reductions, material changes, stuck valves, or any other potential restrictions.
Best Answer
Looks like Landscape wire.
It's low voltage cable (150 volts), so there's likely a transformer and/or control box at the other end. Though the previous owners may have taken all the landscape lighting (leaving the rolled up cable), and possibly the control module as well. So you might just find where the control module used to be.