Control my air conditioner with a Nest

I have a window air conditioner. It works well but is a bit old — new enough to be electronic, have a remote and a digital display, but not new enough to turn on from my phone or do anything smart with its thermostat.

I'm dipping my big toe into electronics by hooking up a Nest smart thermostat to this air conditioner, and it's proving to be an adventure:

*The air conditioner's electronics run on 16V DC. A 68k microcontroller switches the negative sides of some SJ-S-112DM relays via a darlington array on an IC to control the fan and compressor. I'd like to bypass everything except the relays (but leave the rest functional if I want to go back to basics).

*The Nest wants 20-30V AC (24V is standard for residential HVAC). It switches the hot side of the HVAC's relays. It can power itself by running a small amount of current through the relays (but not enough to activate them), or you can connect a dedicated neutral wire.

So, not so compatible. Anyway, I'm a computer programmer by trade, and this is my first electronics project since high school. I'll take an elegant solution over an easy one — especially if I can learn a few things along the way.

I assume that the Nest uses s-state relays to pass voltage out to the air conditioner's compressor and fan. I further assume that that path is agnostic to DC vs. AC. I even further assume that some kind of rectifier is used to power the Nest's guts, and that it will be equally happy on DC.

I've partially tested the last assumption by hooking the Nest up to the 16V out from the air conditioner. I thought that maybe it would just work — that 16V was close enough. Alas, it refuses to operate the system, giving me a "power wire not working" (E24) error. But it didn't blow up! And it even shows the correct voltage on its debug screen. Encouraging. DC might be fine.

Here's my current plan:

*Buy a small DC/DC buck converter to boost the 16V coming out of the air conditioner to 24V. Connect this to the Nest. Its outputs will be 24V, too, though, so…

*Buy two (or 200, turns out they're super cheap) 2N2904 transistors. Connect the Nest's outputs to the bases of the transistors.

*Connect one 2N2904(www.kynix.com/.../2N2904.html) to the compressor relay, and the other to the fan relay.

So the Nest gets the 24V it wants (but DC instead of AC), and its 24V outputs drive the transistors. The transistors switch the negative sides of the relays.

Here's a schematic. It's my first so be harsh with your criticism:
<img>i.stack.imgur.com/.../img>

The questions I could use help with are are:

*Is it safe to drive the Nest, which expects 24V AC, with 24V DC? If not, why not?

*If the Nest expects to switch AC, will it switch DC too?

*The relays don't seem to be specced to use much current. Is it okay to switch them with transistors like this?

If you see any other big holes in my story, I'd much appreciate you pointing them out too.