I am done running means: I have finished a run/jog. OR I am ready to face an issue and won't avoid it any longer. I am ready to deal with something I have been avoiding. I am done with running means: I quit running, I no longer run and don't plan to run again in the future - retired from the activity of running. OR as above, a more slang version that you are ready to deal with an issue you have been avoiding.

You are correct. "I am done *with* running" is proper. When it is shortened to "I am done running" it is correct, but it is colloquial.

Hi Lyndon. What about 'I'm done eating"? Is this an idiom too? What part of speech is "eating"?

Ok first of all in normal English: you'd maybe use "I'm done with running" in the context of you actually taking a run like you would do in the morning or evening routine. "I'm done running." is actually an idiom, meaning "I'm done avoiding the issue." also saying "this is where I am staying/standing my ground, defending my position." - So it's basically (another Idiom) "facing the music". Context is everything with and idiom. I hope this makes sense.