When you press the remote control to open or close the doors, do the indicators flash?
The fuse that powers the door locks in on the internal block at the end of the dashboard on the drivers side (Fuse 14, 15A, helpfully marked "door lock", bottom row, 4th from the left), removing it removes power to the door lock motors. Pulling any of the anti-theft system/ immobiliser fuses ( or them being blown) will of course immobilise the car, which is not the case.
However the door lock controller is also linked to the anti theft module, so in normal operation, pressing the key buttons will open and close the doors AND set/unset the alarm, because the anti-theft module activates and de-activates the door lock motors. This is why if you close the doors with the remote key, but then open them with the non-remote key, the alarm will go off, but if it was locked with the manual key, it will not. Try the above sequence.
No central locking, but flashing indicators and working alarm are indicative of a failed or missing Fuse 14, the wiring diagram makes this very easy to see.
The Aveo central locking system is VERY sensitive to low battery voltage (the main vehicle battery). Even if the car still starts fine, a low battery, (think 12.5v rather than 13.5v) will cause the central locking to become very "chattery" or fail to lock at all. In these circumstances, pulling the fuse 14 will allow you to carry on using the car and alarm albeit using the key manually.
If you only just got the car, you should check the main battery voltage, check the fuse, and check the sequence above for any of us to be able to help. Note that the original battery fitted at the factory was "Delkor" battery, mine lasted 11 years and survived 2 lots of long term storage, it was still capable of starting the car, but was changed due to the above problem.
Check all the above and report back.
Obviously, your next port of call would be the door lock relays......but you say that the central locking works when the manual key is used?
