C wire is the common. Just because it wasn’t attached to the existing T-stat doesn’t mean it’s not there. Pull the wire out from the wall behind the T-stat and see if they cut the C wire off short. If there’s enough slack, you may be able to use it.
Btw, ecobee tech support is very helpful.
Check with the isp for compatibility. I’ve had very good luck with the Netgear CM1100. Modem only. I run a separate mesh router system which allows me to change internet providers or upgrade my modem without having to mess with any smart home devices.