You can use either one. In a formal or official setting, you would specify 'one'. Same goes for other periods of time (hour, minute, month, year).

In other contexts, the use of the word "one" can emphasize that there was one, NOT two, three, etc. Compare: "There was a man in the room." This tells us that a man was there, but doesn't give importance to the exact number. "There was one man in the room." This stresses the fact that there was ONE man there, not another number of men.

Either is fine. If you are "counting", ie. one month ago = four weeks ago = approximately thirty days ago, then it makes sense to use "one". We're usually not so fussy in daily speech, so "a month ago" might be our first choice as native speakers.

Yes, you can use either one! Saying "one month ago" is more specific, but either is fine :)