Is it ok for a water heater pressure relief valve to be moved away from the tank with a short pipe

The short answer: you should replace the relief valve.

The long answer: In general, you should not attempt to repair any pressure relief valve. This valve is the only over-pressure safety in the system. This should only open if there is a blockage in the system (causing a pressure buildup), or if filling the system with a faulty pressure reducing valve.

Normally these valves fail due to calcium build up on the interior, which is hard to avoid without having a water softening system installed.

I would recommend changing out both the pressure relief valve, and the pressure reducing valve, as it appears to have quite a bit of calcium build up as well (and may have a small leak). To have both replaced would generally cost in the range of $200 to $500, depending if you replaced them yourself or not. If there are no other leaks or issues with the system, changing both of these valves will solve your zero-pressure issue.

I don't think that you need to turn off the gas to the hot water heater. I believe that it is likely acceptable to leave the appliance on the 'pilot' setting.

I would obtain the manual for this device (try searching online, you will likely find it that way) or call the support number for the manufacturer.

However, what is acceptable practice is not necessarily best practice.

Personally, I would be hesitant to leave a gas appliance unattended for more than a day or two without physically disconnecting the gas (usually a 1/4 turn valve on the feed line). However, I feel comfortable lighting the pilot light myself.

Perhaps your should check with the local utility that provides your gas and see if they offer a service to visit your home and relight the pilot light upon your return? Mine does and it is free. My wife knows to contact them in the event I am unavailable because she is not comfortable at all in doing this herself.