Okay, now we're making progress.
First: Of course how you hook up the nano matters! The electrical system is the way it is for many different reasons. Taking power directly from the BEC will cause your robot to constantly reboot as minor voltage sags cause the nano and HC05 to brown out. However, this is probably not the cause of the particular problem we're discussing.
Second: The "curling up and dying" behavior can be caused by a bad BEC. Are you using the exact BEC we use? (A picture is in the instructions). If you've substituted, you could have a bec that claims to output 3A but really dies much sooner than that. We have a list of BECs that are known to be bad on the circuit diagram page for self-sourcers.
Third: If you're detecting a short when V+ and VCC are connected on the servo driver, then either your servo driver board is bad or one of the devices connected to the servo driver is bad (or wiring is wrong, but let's assume you've double-checked that.)
One candidate is the buzzer, I've seen that happen. However, you are now reporting that the buzzer is working, so that's not likely
If you've got the ultrasonic rangefinder connected, disconnect that and try again. If everything works, that was the issue.
If you have a second servo driver, you could swap that in. But in my experience, that's rarely the issue, so let's try other things first if you don't have a spare.
If none of the above is the problem, then that leaves just the servos. Cheap servos often do have defects and some of those defects can cause a huge current draw.
You could disconnect every servo, then connect them back one by one. If the first one works, put in a second, a third, etc until failure occurs. If the first one doesn't work, then disconnect it and add a different one. By doing this, you'll figure out which servo is the problem in 5 or 10 minutes.
Hope this helps!