2005 Chevy Cobalt 2.2 liter Automatic will not go into reverse. Drives forward great????

Ok, well nothing electrical can cause a no reverse condition. There is some sort of internal failure. Many things can cause this.
From the reverse clutch, to the low reverse band, or a servo piston, or stuck check valve, manual valve, etc.

Answer: It is going to have to have the pan dropped and look for dirt, then most likely be removed and disassemble to look for the failure.
I looked at some TSB service bulletins, and found one for a failed reaction sun shell, which may also causes no 4th gear.

Chevy Cobalt BCM Replacement

Problem with my = 2008 Chevy Cobalt. 2.2 Ecotec. Check engine light.
Replaced BCM.
I have faulty BCM can it be replace with a used one without having a gm dealer reprogramming it.

Answer: No, you cannot use a used BCM without it getting programmed. It most likely will not start.

The theft system would need to be programmed, as well as other vehicle options such as steering, airbags, lighting, radio, etc.