Generally majors go to one of the Command and Staff courses. A LtCol select might get war college, especially if he was scheduled to pin on during that year, but otherwise, I personally don't know of any Marine majors at any of the war colleges, not to say that it's never been done.
There are some special programs allowing study at civilian universities. Most of them are fellowships for senior officers. There's a billet for ltcols at the JFK School of Gov't at Harvard, for example. The only one that springs to mind for company grade officers is the Olmstead Scholarship for studying abroad. Suffice it to say, these are pretty rare.
You can put in for special programs at any time. The usual career gouge is to do it right out of a fleet flying billet, so as to minimize time out of the cockpit. Doing it next to a FAC tour would be doable, but I'd ditch the FAC thing altogether if I was doing grad school.