I’m a mechanical engineer (Ph.D. UIUC).
On the question about CS vs: Yes. This is very important to mechanical and all types of engineering. Take the programming course if you can.
If you take AP Physics C, take it if you can. That said: I would advise retaking the introductory in college unless you get well above 5’s on both tests. Owing to the generous curve, those who get a single 5 just skimming above the cut-off are definitely NOT for follow on courses in solid mechanics, fluid mechanics. AP C doesn’t cover thermodynamics at all. This is very important topic in mechanical and chemical engineering. This is often covered in introductory physics for engineers and physicists-- but is not on the AP Physics C curriculum. (These sorts of mis-matches are a reason engineering schools often won’t even accept 5s on the AP C when granting credit for introductory physics. )
Going forward in ME, you want to be rock solid on mechanics, thermodynamics and it helps to have seem some fluids. (Fluids is also absent from the AP Physics C curriculum. That said: it doesn’t matter that much. The first fluids classes in ME and Chem E pretty much assume you didn’t learn any in physics).
You already look good on math. Did you get a 5 on the earlier calculus AP exams?
As proud as kids can be when they get 4s, if you are going into engineering, you should figure out a way to retake these foundational math courses if you got 3 or 4s in those math courses. (Your university engineering department l may require retaking them in any case).
Content in the first three calculus classes re-appears in thermo, circuits, mechanics of materials, fluid mechanics and heat transfer. If you got a 4 on the AP test, you will find yourself in classes where faculty expect your mastery to be better than the level reflected by a 4 on AP tests.
For what it’s worth: calculus, linear algebra and advanced math becomes very important as you advance in economics. So you want a firm foundation there, but I don’t know the specific expectations of econ faculty.
On another issue: going forward in college, your college GPA will matter when people or agencies are considering you for scholarship which sometimes come up, college awards, college honor societies, admission to grad school and so on. You will almost never find anyone who will give “atta-boy” add on consideration for you having APd out of the entry level course. If they are important to your major, retake those courses unless you are absolutely, positively rock solid on them-- meaning you would have gotten an at least a B+ on the previous years final exam at your university. You want to master material in your major to the A/B level at least.