1. No clue, but as engineering friends have told me many times over, 1 to 2 years after you graduate, no one will care where you went to school.
2. Math and engineering classes go hand in hand and being strong on verbal problems is very helpful. One of the common things I see in my classmates that are struggling is a failure to recognize when they are expected to know a certain formula and when they are expected to recognize that a certain problem requires an integral or diff eq. That being said, doing well in pre-calc is good, but it does not prove that you have the kind of brain that can get through higher math, but most likely you do. I'm currently about 2.5 years into my engineering program and I just finished my last required math class. Engineering students can generally be put into 3 groups when it comes to math.
Group 1 could easily be math majors if they wanted to be, they choose engineering because of money, interest, or any other of a thousand reasons. BTW, I am so NOT in group 1
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Group 2 is where I would place myself, I'm reasonably good at math. I don't like pure math classes but I get the material well enough to pass on my first try. I know that the important material from those pure math classes will keep popping up over and over again in my other classes, so I don't sweat getting a C+ or B- in my math classes. This is the same group that I would place most engineering students. For this group, getting an A or B+ in calc I required little effort. Getting an A or B+ in calc II was possible, but would have required a LOT of extra study time that may not have been available due to other classes. Getting an A or B+ in multivariable calculus, diff EQ, and linear algebra would have required more time then most group 2 engineering students have available or might be impossible due to reaching the extent of their brains ability to understand mathematics. In othe words, if you put in enough effort, you can get through the math, your just not a math wiz like group 1.
Group 3-These are the engineering students that are taking calc II for the third or fourth time, and not just because they wanted to improve a C up to a B or A. Taking the rest of their math classes will be pure torture for them and they may have to repeat other classes multiple times. If they are persistent enough they may get through an engineering program, but it will be a huge challenge.
Sorry about the huge rant on math, but trying to discuss engineering without talking about math is pointless. Basically, if you have to take 1 or 2 math classes over again because you dropped them or failed them, you won't be the first engineering graduate that repeated a couple classes. if you have to take most of your math classes 2 or more times, you might be in trouble as far as engineering goes.
3. I'm about the same. I hate walking into a lab without understanding every line of code (if it's a programming type exercise), formula, and piece of equipment that is going to be used. Needing to understand things made some of my math classes more difficult then they should have been. Sometimes proofs of certain formulas were not given because the level of mathematics required to understand them was "beyond the scope of this book", I truly hated seeing that quote. It comes down to the difference between knowing something and understanding something. It's like the difference in KNOWING that 5*5=25, but not being able to apply it to the 7*3 and UNDERSTANDING that 5+5+5+5+5=5*5=25 and then being able to do 7*3=7+7+7=3+3+3+3+3+3+3=21 even though you never saw 7*3 before.
4. Don't worry about what you may have missed in HS, the important stuff will keep popping up over and over again.
5. Designing is problem solving.
EDIT:
One last thing, don't worry about being an older student, I was 31 when I went back to school to get an engineering degree.
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