Toilets are typically supplied using 1/2" pipe, while it's common to use 3/4" to supply a spigot. You can certainly tee off the toilet supply and feed a spigot, though you might not quite get the flow you're expecting. Should be fine for watering flowers, vegetables, etc. Though may come up short if you're trying to water the lawn with a sprinkler. You'll also probably notice a pressure drop off, if somebody flushes the toilet.
If you end up supplying a spigot (however you do it), I'd recommend installing a new anti-siphon frost proof sillcock. The one that's there looks like it's seen better days.
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
The washer that seals the water flow may have become unscrewed from the end of the shaft. I had this problem. I had to remove the shaft and tighten the washer back on the end