Not exactly. USAFA is your best chance at getting a pilot slot if you want to talk strictly about odds. Put it to you this way, the Academy by itself gets about the same number of pilot slots for one class that the whole country's ROTC class gets. So if you want to all but guarantee a pilot slot, go to USAFA. HOWEVER- once you get to UPT, you're competing on the same level playing field as everyone else- ROTC, OTS, and USAFA. Your diploma from USAFA isn't going to help you or give you a leg up on your peers in UPT who are also trying for a fighter slot.
Since your goal here is to become a pilot, your degree won't really help you or hurt you. You'll take enough core engineering classes at USAFA regardless of your major that will "shape your mind" around techy subjects, so major in whatever you like. Everyone graduates with a Bachelor of Science, whether you major in Philosophy (and yes, that's a major at USAFA now) or Physics. That being said, if you have any aspirations to become a Test Pilot at some point in your career, you'll need to have majored in engineering. So keep that in mind when choosing...
I knew a few homeschooled dudes there and they didn't seem to have any advantage/disadvantage. You're either a bro to your peers or you aren't. Be humble, help out your classmates, and don't take things too seriously.
If you plan on trying to make the Flying Team, finish your PPL. I'm not saying you can't go to USAFA without a PPL and still find the time to get it before Flying Team tryouts, but I'm saying you can't go to USAFA without a PPL and still find the time to get it before Flying Team tryouts. And if you're like me and didn't find the time to finish it before I went, feck it. Try out for the Wings of Blue instead. As a former team member, we had the most fun of anyone at the academy, and no one will ever convince me otherwise. Ever.
At this point, you're getting a little ahead of yourself. First, manage your expectations. Dropping a raptor or strike eagle out of UPT isn't just about how bad you want it or how hard you work. Timing is everything. Go check the most recent UPT drop out of Vance on another thread on this website, and you'll see that only one person in the class got a fighter which happened to be an F-16. When you graduate UPT, you're at the mercy of the needs of the Air Force. If they don't need any raptor pilots at the moment, then no one in that class will drop a raptor, even if the number 1 graduate listed F-22 as their number 1 pick. That's just the way it is. But to answer your question, start developing a solid work and study ethic. Lame answer, I know, but that's what will get you through any rated training program better than anything you can study or work on now.
If you're talking about percentages, the Navy has the highest percentage of fighter pilots. So decide if you want to live on an aircraft carrier for a significant portion of your life. No thanks...
Is it possible? Yes. Should that be your goal from the start? No. Every service academy class has a few of their graduates that cross commission in to a different service branch, but I wouldn't bank on it.
If you drop a heavy out of UPT, you're flying heavies for the rest of your career. Have there been exceptions to that rule in the history of the Air Force? Sure. But those instances are few and far between. So the straight answer to that is no, you cannot cross train to fighters after entering the heavy world. That being said, don't knock it before you try it. I have plenty of friends that fly something starting with a "C" that wanted fighters originally but that now love their aircraft and are glad they fly it.
Hope that helps man. You're welcome to PM me if you have any other questions. I don't come close to having all the answers and much is to be gained on this forum from folks wiser than me, but I'll pass on whatever I can to help you out. We've all been there in one way or another...