As I'm doing a physics and maths degree, I'd say that maths is pretty important to me. I'd also like to believe I'm reasonably good at it (especially the applied maths side of things, the hardcore theory goes over my head usually).
I was also horrified to find out that a friend of mine has a girlfriend who got a maths degree in the US, where she didn't even go beyond what we do in the start of second year maths here (no multivariable etc.). That's just... how can you even call it a maths degree?
As for order of operations, I was taught BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction) with the understanding that Division and Multiplication are commutative, as are addition and subtraction (because they're essentially the same operation, dividing by a is the same as multiplying by 1/a, subtracting by b is the same as adding (-b)).
I do think the order of operations is important, even if it's entirely arbitrary. This is how you get consistent results. Without it, we're in trouble.