If the batteries in a thermostat are almost dead, will the furnace not work properly?
No, the furnace would not work correctly. The batteries serve 2 purposes. The first being to power the display so that you see the current settings on the stat and to just them accordingly. Second (and more importantly) the batteries power a relay that sends low voltage (24 vac) back down to the furnace to signal the circuit board to engage the desired services. If the batteries are not providing enough power the relay will not remained closed. Especially for air conditioning, where 2 relays need to be engaged, 1 for the condenser (outside) and the other the fan (inside).
What does auxiliary heat mean on Honeywell thermostats? How can I fix it from running on emergency or auxiliary?
"Auxiliary heat" means the secondary heating source. You have the thermostat set to keep your maximum temperature at a certain point, at which the device providing heat turns off. Your thermostat has two possible heat sources, a main and an auxiliary. It depends on how the unit was wired which heat source is which. For example, you may have off-peak electric as your main heat source, and a gas-fired furnace as your auxiliary, a wood boiler as your main and a heat pump as auxiliary, etc. Normally, the auxiliary will only engage if the main is not functioning, but there are a host of options. It may be set to auxiliary. You can get your thermostat's manual as a download from Honeywell. You need to do this and read the exact instructions for your exact model. Without that, an answer is a mere guess.