2005 chevy malibu - remote starter stopped working. When buttons are pushed, lights flash, sometimes engine turns over but never starts. Door and windows all open, don't think battery is low. What about receiver in the car? Not getting correct signal?

Is the remote starter factory or aftermarket?

The starter is factory installed.

Do you encounter any problems trying to start the vehicle with the key? Long cranks, hard starts or stalls?
Typically key Fobs/Remotes use what's called an alternating algorithm to relay signals to the vechicles ecu (engine control unit) and/or signal receiver. This algorithm is a series of numbers and letters that change in a pattern every time you push a button on the remote, the vehicles ecu is also rotating numbers in the same pattern. This is why your remote is the only one that your vehicle recognizes.
If the engine turns over only occasionally then you may have an intermitten problem with a relay or switch. The easiest solution would be to take it to the dealership, however if your attempting to resolve this on your own I would recommend starting with the simplest solutions.
1. Check the cranking amps in the vehicle battery and replace the batteries in the key fob/remote.
2. Check the fuse panel for any blown or missing fuses.
3. If you have a check engine light on use an OBDII scanner to check for codes.
4. If you find any codes let me know, if you find no codes then proceed to check all the wires for any shorts or loose grounds, use a voltometer to check for continuity.
5. Continue to check any relays or switches that control the remote start.

Remote starts are typically programmed into your vehicles ecu and have a set parameter for cranking time and tempature conditions, it may be possible that your ecu has encountered problems with these parameters, this would require diagnosing by your local dealership.