Open Engineer Mode and run other hidden codes on Moto G 3rd generation device?

These dialer codes all vary by OEM in their stock ROM's... the codes for stock firmware are often easily accessible via searching the Internet or via apps available from various sources. However, how custom ROM makers handle these codes, if they implement them at all, is an entirely different thing again.

There are some standard-ish codes, like *#06#, but even this is not consistent across all devices in all cases. There isn't really a standard for other codes nor is there any incentive for OEMs or ROM developers to even implement many of them.

Many dialer codes are also passed to the carrier, for things like Call Forwarding, checking your balance of money or minutes, blocking Caller ID, activating Call Waiting, etc... although there are some standards, actually commonality more than a standard, between some carriers, these are entirely carrier dependent as well on how and if they implement them.

To answer your questions specifically, well, as specifically as possible...

I would like to know if there is a way of looking if the relative
services are on my phone? If it isn't the case, how I can add them or
run from PC with USB maybe?

Unfortunately, no... these are device and ROM dependent. I would suggest posting in the official support thread for your device and specific ROM to see if they have implemented in any way or there are alternatives that are specific to your ROM, device, and use case.

Are the secret codes dependent on the ROM like I imagine or not?

Yes, they absolutely are...

Source: Years of being an Android enthusiast, ROM developer, and XDA Recognized Contributor. :)