Does T200 thruster along with the BESC30-R3 draw any power when no PWM signal is provided?

No - I was testing idle current draw of 5 different ESCs, to try to get some idea of the variation between them.

From my (admittedly limited) measurements, it seems that multiple ESCs when powered together by the same supply use a bit more current at idle than they would when powered individually. I’m not sure why that would be the case, but I don’t expect it’s caused by a very low resistance connection between them (like the barrier block) - I found similar behaviour when I just powered two and held their power wires together (If I’m remembering correctly I think they were drawing ~9 mA each).

Fair enough, this comment may be worth a read. Note that getting a single relay with sufficient power capacity to handle all your ESCs may be challenging (depending on your operating requirements), so it may be necessary to have an individual relay (or MOSFET) per ESC (perhaps with a shared switch signal).