The problem is your expansion tank, or lack thereof. Your home has a check valve somewhere in the system, possibly included with the pressure regulator or as part of the municipal water meter. When your water heater turns on, the heating of the water causes it to expand, and that pressure has nowhere to release until you open a faucet. What you need is an expansion tank installed in your water lines. If you already have one, then it has failed (they usually have a bladder separating the air from the water, and this can fail over time or if not properly maintained). Installing a tank isn't too difficult, but will require shutting off the water to your house, cutting your pipes, and adding a T to your water line. If you have an existing tank, then they are often threaded and are simple to replace (the connection is often threaded and can be screwed off). Note that if you don't fix this, valves in your toilet and other fixtures will start to fail, or the TPR valve on your hot water heater will begin to leak water.
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
I wouldn't risk connecting the rain barrel to your house's plumbing because you don't want to risk getting non-potable water into your pipes. This sort of arrangement might even be illegal where you live.
What if, instead, you just use a garden hose to add more water to the barrel as needed? Then you don't need to worry about running out. Or, use a quick connect fitting to make it really easy to disconnect your watering hose from barrel and connect it to another source.