Aside from the function/spatial requirements, you are not limited to the 2x4 or 2x6 stud dimensions but your top plates might need to be a little more stout if you are going to support multiple floors. The header functions in all purposes like a beam between the studs (which in all truth are tiny little "columns" or "posts" if you may. Keep in mind the load you are applying on the "bearing" header and the columns. Most likely, it's not the studs that will be the issue but the double-top plates of 2x4 or 2x6 will max out sooner in it's load but for up to 3 stories, it might not be that much of an issue unless you are using some unusual load but if you may, you can change from a double-top plate to a 4x6 or 2x6 to 6x6 header beam. You may yet still use a 2x4 or 2x6 on top if you want to be technically a double-top plate but that would be unnecessary. If I was designing a 2-3 story home with 24" o.c. then I would change the top header into a singular solid beam that would span farther. You mentioned thermal bridging so you would have to account for that where the roof, wall and ceiling joins. If you are platform wall framing, you have double top plating at each floor and each platform framed wall is one story tall and essentially stacked on top each other for multiple floors. In a balloon framed system, this may be different so you have to account for in some fashion in the structural design and but running 3 story structure on regular 2x4 or 2x6 stud walls at 24" o.c. would be sketchy, in my opinion but on a 1.5 story structure or a 2.5 story home.... not a big issue especially if the walls are just one to 1-1/2 story tall stud frame from middle floor up to about say 5 feet above the upper attic/loft floor line while the ground floor being a solid concrete/CMU or some other more "sustainable" floor system that is adequate for the load. In any case, there is a lot of options. Some options might need an engineering consultant but sometimes, it doesn't. I don't think 24" o.c. spacing should be a problem for a 1-1/2 (1.5) story home but when you add multiple stories, that can be a problem as the cumulative loading may exceed the load bearing capacity of the studs which can handle up to a certain amount of load (axial load capacity). Basically, you can compute that as you would any post or column. What you don't want to do is overload those little "columns" or they will start potentially splitting vertically.
24" o.c. isn't necessarily the problem. Depends on the load. If you have concentrated loads that needs to be supported, it might be on the weak side with regular nominal 2x4 or 2x6 studs.
The header should span but I would probably beef it up to a single 4x4 or 4x6 ( with grains flat in the best orientation for use as a beam ). If your wall is a 2x6 wall, the 4x6 should be flat so the. The extra 1/2" of vertical depth would be adding a little more load carry capacity to span 24" but you could go to 6x6 if you feel more comfortable with spanning 24". If you're wall is 2x4 walls then 4x6 on edge (with grains flat in the best orientation for use as a beam) would be more than adequate but you'll need to plan out right where the header will be before the sill plate of the next wall up (assuming you're using conventional platform framing) to address thermal bridging. In cold environments, I might actually go to 2x8s as studs because it provides a deeper wall cavity for more insulation value.