I think my grandmother might have said she didn't give a hoot about something. I've never heard the other two expressions the way they're written.
I've heard people say they don't give a damn about something. In your example you wrote "curse" and yes, there are several curse words that people don't give about things: a s***, a f***, a rat's a**, a toss (British).
Not giving a damn is quite dismissive and already a little bit rude (with ruder options available), so when you aren't looking to be offensive you can say you couldn't care less about something.
You will post your project when you're done with it, won't you?