Looks like you have been having fun . When the voltage can change as it does down round the levels you are working at ,any calibrations of devices read by the arduino can be out ,by 10 degrees C in my case. The arduino (atmega328 in my setup) needs the voltage to be over about 7.4V it seems to hold steady voltage in.
I have a 12V SLA 7AH battery attached to a small solar panel .It drives a couple of 1 watt LED lamps at night . Hooked up the arduino and it runs fine. Monitoring a freezer . Only had it running for about 2 hours and had to rework something else on the freezer controller so turned it all off .
Didn't see any reason for it not to work for long periods. Arduino is consuming about 40mA as I have it set up . The voltage regulator was holding steady at about 4.93V and that should continue down to about 7.4V out from the battery and I wont let it get down there.
I am working towards radio modules on the freezer end and on the computer next.
I like the LEDs under the table though ,might put some in my setup which will also light and run a fish pond at night. Thats another story