Sanity check my first project: to control my air conditioner with a Nest
If switching relays with NPNs, you'll want a diode to protect against coil reverse-induced emf when it de-energises, as per the diagram below. 2N2904 should be fine. Also, I would definitely use a resistor at the base of the transistor that would limit the current into it to somewhere between 2 and 10mA. I have assumed the next outputs a positive DC signal at Y1 and G at 5v DC.





As for your other questions:


If a mechanical relay is capable of switching 24V AC, then it will be able to switch 24V DC.
If the nest expects 24V AC to power it, it is not possible to say if it would run off 24V DC without seeing its schematics. While I suspect you're correct, especially given your test at 16v DC. (However, I would have been hesitant to try it).