What was Carolyn’s intention with the gun?

This scene is set-up as sort of a who-dunit to start with. The actual scene where Lester is killed just shows a gun to the back of his head and it's unclear who pulled the trigger. His daughter looks very serious in the video her boyfriend takes of her when she says she wants her dad dead. Her boyfriend then gives a very serious ( I might do that for you) look back. Lester has also just ruined his wife's affair and her business partnering which are probably the two most important aspects of her life at this point. Plus with his change in behavior lately she is very jealous of his carefree life but can't let go of her image of who she is. So with everything seemingly falling apart and considering her violent outburst before I think that there is a very good chance that she planned to threaten him with her gun.

She doesn't have the guts to actually kill Lester but she is mad and her world as she knows it has collapsed and everything can be traced back to Lester changing on her. So yeah she would want to threaten and scare him, but I think you are correct she wasn't really going to kill him. She's just a traditional "red herring".

I'm pretty sure Carolyn intended to kill Lester, or at least she believed she intended to. She had been listening to the "take life by the horns" motivational tapes and felt she was justified in removing Lester from her life. She looked very intense sitting in the car before going inside.

It wasn't until she heard the gunshot and found Lester dead that she realized how deeply she loved him, as evidenced by her melting into his clothes while crying hysterically at the end.