I believe the left pipe is the hot water. If I were to turn it off, which valve do I use? The blue valve or the smaller chrome one on top of the blue valve? What are the differences between these valves?
If I turned off the hot water entirely for a few weeks, are there any concerns about doing so? I would just use the cold water in the sink for the time being.
The left one would normally be the hot water and the additional tap goes to your dishwasher. Either of those two valves on the left will shut the hot water off to the faucet and the dishwasher. Both types serve the same purpose but the smaller chrome ones look better. I would guess since you have copper/brass drain pipes that the original supply lines to the faucet were copper too. When the faucet was replaced, the installer went with the braided supply lines and new valves that wouldn't require sweating any copper and left the old valves rather than remove them.
Turning off the water for extended amounts of time will not affect them.
If you have all four of the valves shown, in the open position but you can’t get warm water at the sink faucet, only either hot or cold, then there’s probably a problem with the sink faucet, assuming you have a single lever type faucet.
The two top valves accomplish the same thing as the two lower ones.
My guess is that the lower valves (blue, black handles) are old and don't seal fully. The "plumber's" 'fix' was to install riser pipes to the second set of chrome shut-off valves rather than mess with re-packing or replacing the lower valves.
And I put "plumber" in quotes for a reason. It looks like they messed up the valve install. They didn't leave enough room to turn the valve. Notice the screw? It's been backed off so that the handle can be removed; they must have used pliers to turn the valve instead. That's just... lame.
An answer then. Take the handle off and use pliers to turn off the left-hand valve. Leave the lower ones alone. Consider getting a plumber out there to fix that mess.
As far as leaving it off for an extended time, there's no issue.
As others have said, you are probably correct that it's the left two that matter. The top valve looks newer, so use that if you can. If it's easier to turn off the bottom valve (because the valve is too close to the side to turn), go ahead and try that. If it doesn't leak, things should be fine.
You also asked why there are two valves and what the difference is. It looks like the top valve converts from 3/4" pipe to 1/2" pipe. The bottom valve looks like outlet and inlet are both 3/4". This makes no difference to valve functionality.
This is speculative, but may be as good as you'll get without asking the person who did it. My guess is that the new fixture takes 1/2" pipe. This is strongly preferred for water conservation reasons. Your original pipes are 3/4". While it is possible to go from 3/4" to 1/2" without a valve, they probably find themselves frequently adding valves that do this. The cost difference is minimal, so they may only carry 3/4" to 1/2" conversions as valves. In the end, it was just easier to add a redundant valve than to wait for the correct part.
Other speculations may also be true, but I think this is the simplest possibility.
I agree with those who say that you can turn off the valve for extended periods of time.
