Sure - however, tuning is going to prove to be a bit of an issue.
The '05-16 Cobalts used an older 16-bit computer that makes it very difficult to tune (and virtually nobody will tune one). I think people have better luck with an aftermarket/piggyback computer, though, but I could be wrong.
Of course it’s possible. But you are unable to tune the 05-06 2.2 PCM. The stand alone systems are what have to be used for fuel control. Probably would need a custom ECM. The later models used a 32 bit PCM. I believe the 2.0 and 2.4 were already using the 32 bit system from the start.
About all you can do on these is install some bolt ons. Header, cat-back, intake mods. You can however can swap many suspension parts and brakes from SS models.
I didn’t buy my car knowing it’s limitations. Found that out a few months afterwards. But it is a cheap car to make nice.
Even the engine swaps in these cars is ridiculous. All wiring and modules would need to be changed to make things work. At least from what I’ve read over the past year and a half. Not an easy swap like an old Honda or something like that.