AIUI GV is not able to forward to specific Callcentric extensions. You can make GV set the Caller ID on the forwarded call to Callcentric be the GV number instead of the original caller's number. Then a Callcentric Call Treatment can act on that to ring the appropriate extension. However that means you have neither Caller ID nor Caller Name of the original caller.
So I think you need another DID (ie real phone number) either added to your existing Callcentric account or in a second Callcentric account.
You might instead consider using the Simonics GV Gateway when it is open again to new subscribers. A one-off payment of $6 versus the monthly 911 fee to Callcentric.
The distinctive ring bit is straightforward. Each Voice Service can select either A or B Ring Profile and each of those profiles has 10 Ring Patterns.