As long as the attached timer is shut off and no water is flowing the pressure in the water line will be no different than what it would be when the spigot is turned off.
You water bill will reflect how much water that you actually use. The only way it could go up astronomically would be if the timer broke off the spigot or failed in an open mode where water was allowed to flow freely from the spigot for long periods of time.
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
Closing the inside main valve to your house should do the trick. If you haven't operated the main valve regularly do so with caution because gate valves are noted for failing when you close them. After doing so, check any faucet to verify the water has been shut off. If you live in a freezing climate you probably have an inside shutoff valve close to the outside spigot that you cold use instead of the main one. After closing the main valve, if it' a gate valve, you might find that it won't open back up and that the handle just spins. That's due to corrosion and the valve has to be replaced. If you have access to the street cutoff and the water authority has no problem with you turning it off there, that would be a good option. They could possibly turn it off for free while you change your spigot.