Electrical – Turn on gas wall heater w/o electric thermostat

Many thermostats make light(er) work

What you want isn't a single central thermostat. On the contrary, a key principle of proper building control is that parameters in a zone should be controlled by a controller in that zone, reading sensors local to said zone, and driving devices that control nothing besides that zone.

This means that the devices you want to install are individual thermostats, but with networked supervisory capability -- i.e. the ability to set the temperature and scheduling over a local network from a central point. This sort of use-case is where networked thermostats (there are several out there) excel; however, many of them are designed for more traditional 24VAC control systems used with furnaces and heat-pumps, not the line-voltage system used with electric wall and baseboard heaters like yours.

There are a couple of networkable line-voltage thermostats out there; most notably at the time of this post, Honeywell makes one for their proprietary RedLink wireless automation system (albeit without any type of API integration) and Casa makes the WiFi based Caleo. In addition, existing low-voltage thermostats such as the Nest can be used with line-voltage systems using a relay transformer combo such as the Honeywell RC840T or equivalent, although the RC840T's transformer may not provide enough power for some networked thermostats (such as the Nest).

Whatever system you go with, though, make sure you secure it from outside network access!

It sounds as though there is a short in the wiring somewhere. Most fireplace remotes will still function manually with the batteries removed. The C34 utilized a SIT millevolt gas valve. The remote controls' 2 wires will be attached to the valves "TH" terminal & the THTP" terminal (top & bottom terminals). If you remove 1 of the wires & the issue ceases, its a remote problem - & the fireplace is fine. At 17yrs old, its probably the remote. The reason that the burner comes on 30 seconds after lighting is because that about how long it takes for the thermopile to generate enough power to energize the valve. The click you hear shortly after shutting off the gas is the control valve closing from the pilot being shut off - a safety device.