Approximately one-third of the graduating class in recent years goes active duty. The specific number of slots available in each branch varies from year to year, but I have not heard of anyone wanting to go active duty being denied. Not sure about Army Combat Engineer in terms of likelihood, but I know of one of my DS's classmates (2013) who went that route from Kings Point.
I have. The old army liason told us about how they denied someone and emphasized that it's not what can the army do for me but what can I do for the army... They definitely don't take anyone and everyone. It's needs of the service and your participation in army ops to get the liaisons blessing.
One of my classmates transferred to West Point for this exact reason.
Going into active duty Navy is more seamless because the dept of naval science is on campus and can help with that a lot more easily. But there's no hard and fast numbers on spots. There could be few or many depending on the services target for accession.
My "almost daughter" is at KP...she's going on active duty. She said there was something like 20+ that wanted her branch and only 8-10 were chosen. Those are my numbers, I don't remember precisely. But there were quite a few not selected.
I think one of the other posters said it well: what do you bring them?