I'll address the question in the title: how to get into actually designing an entire aircraft. Not every component, but making the major, visible decisions.
Such jobs do exist, but they aren't easy to get. Design is an iterative process that goes continuously more in-depth on every step. It begins with a requirements document, which are responded to with design proposals, which then get elaborated on for at least 3 steps - conceptual, preliminary, and detail design. Each step takes more than 10x the number of people than the previous.
To design the entire aircraft, or at least a large part of it, you want to be in that select 1% involved in conceptual design. For that, the primary degree to get is specifically Aerospace Engineering. Anything else, more specific, is too likely to pigeonhole you into one of the departments that do detail design work. For general design, where you'll be selecting the planform and the engines, you need breadth of knowledge more than you need depth.
Let's look at three ways to do it:
- Big industry. Seems obvious, as you'll get hired right out of college, but the hard part is getting the right job. New engineers start in detail design, and most engineers stay there. For each person making major decisions, big aerospace needs 100 rank-and-file engineers that just make the 3D models and fill out the part lists.
Working your way up through promotions and lateral moves may work, but runs the added risk of getting detached from the technical work. You'll have to make yourself extremely visible, inside and outside your company, specifically showing your interest and ability to create or develop design concepts.
Most if not all large aerospace companies have a "Skunk works"-like unit, which among other things plays around with pie-in-the-sky projects. That's your goal, you have to get in. That takes reputation, and gaining it while doing your basic duties is comparable to doing a full-time job and a Ph.D. at once - in fact, one approach is to do just that. You'll have next to no free time, but I know people who have achieved this. This is a difficult approach with a slim chance of success, but the upside is that you'll be financially secure in any case; at worst you might be bored.
- Startups. Starting out in as the smallest companies that still design and build aircraft is a more sure way to get to design at least something. It can be a startup that's playing with ultralights, electrics, or just unusual ideas. Such companies avoid or outsource low-level work, defer the painful weight-cutting phase, and focus on the broad strokes.
That will get you to work in a very small design team, building airplanes just as you're designing them, and possibly even flying them. To join a company like that, piloting skills can be more important than an advanced degree, and experience with homebuilts can just win them over - they're looking for enthusiasts, not 9-to-5'ers.
This path is less selective than the other two, but you better have a good plan for supporting yourself, as flight school and homebuilts will be big out-of-pocket expenses, especially if taken while in college (you want to start early, after all). In other words, it's risky. But "full-stack" design experience in a small company can then get you into conceptual design roles in progressively larger companies. Eventually you might end up leading a large project, or even running your own company - startups prepare you for that better than office work does.
- Test pilot. This is less designing, more flying. Being both a test pilot and an engineer is not only doable - a college degree is already a common prerequisite for becoming a military pilot, and aerospace engineering is the perfect choice.
To get from there to being involved in design, you need to be a test pilot. This job isn't about just flying, but about evaluating the aircraft, requiring a lot of fundamental knowledge. Not all test pilots come from a military background, but military flying is a proven fast track. An advanced degree in a related field is also common for getting into a test pilot position, and it will refresh your technical knowledge.
This is less likely to get you into a decision-making role, but it can get you on the principal team, where your experience and input help drive the decisions. In smaller companies, test pilot and engineering roles can be combined. If you maintain and build upon your engineering skills, test pilot experience will also open a lot of extra doors for you - an engineer that knows first-hand how design decisions affect flight characteristics is highly useful at early design stages. Your roles will tend towards consultant rather than staff work.
To sum this up, all 3 paths start with an aerospace engineering degree. Then:
- Flight school -> resume-building (homebuilts, jobs, publications, etc) -> startups (you're there, designing experimental aircraft) -> sell yourself up to design bigger things.
Requires: Lots of enthusiasm, obsession preferable.
Risks: Running out of money.
Bonus reward: The fun starts right away; lots of innovation on the job.
- Industry -> advanced degree/Ph.D -> public visibility -> climb the ranks -> try for "Skunk works" style teams.
Requires: High intelligence, office skills.
Risks: Getting stuck in detail design.
Bonus reward: Big projects, lots of funding.
- Military (officer) -> pilot training -> service -> advanced degree -> test pilot -> industry job. This answers your desire to build and fly planes, with a focus on the flying part.
Requires: Excellent fitness + intelligence + discipline.
Risks: Not making the cut for a pilot.
Bonus reward: Flying the coolest stuff.
None of these is likely to be a straight road. These are just checkpoints to reach; you'll do a lot of other things along the way, building up knowledge, experience, and visibility. But purposefully developing your competencies towards this goal gives you a shot.
As other answers have outlined, there's a lot of different jobs in aircraft design, and they have different requirements. Talented people can get into conceptual design from any starting point in the industry, and you may cross several disciplines. The routes above are just some of the ways to maximize your chances.
In any case, start early. Pick up some light reading, add aerodynamics, build a model plane, fly a sim, get to the real planes once you can. Make dozens of paper designs, each more detailed than the last. In college, ask the most questions, jump on every relevant project, go to every conference, take every opportunity; be a star student if you can, or at least the geek among the geeks. The kind of jobs you're after require as much knowledge about different aspects of aviation as you can get, so use the time you've got.