There are slights drawbacks, but they are barely noticeable. Theoretically speaking, a pure Black Mage will be slightly more efficient than a Black Mage + X Job, but in actuality the difference between a pure Black Mage and a Black Mage + X Job isn't noticeable enough to make a difference.

Both equipped Jobs level up at the same rate and both benefit from the bonus given by participating in an event/mission.

If you take off a Job, so long as you keep it in your list of available Jobs, all unlocked abilities will remain. So in this case, switching back to fighter after three months, you would be keeping up where you left off.