Setting minimum temperature for gas water heater

Such a variation is not normal unless you are using an abnormally high amount of water so that the water heater does not have enough time to recover. Otherwise, since the heater worked properly when first installed, I would recommend calling the installer.

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.