Order of operations - why are they in the order they're in?

The precedence rules allow efficient representation of polynomial expressions in a concise normal expanded form. If you reversed the precedence of multiplication then the notation would be more efficient for factored polynomial expressions. But, generally, polynomial operations are much less efficient in factored form (compare, e.g. equality testing and addition).

The reason that polynomial rings are ubiquitous is their universality (they are free $\rm\:R$-algebras). So, for example, any polynomial identity that you prove true in $\rm\:\mathbb Z[x]\:$ will hold true over any ring, e.g. the binomial theorem.

I don't think there is any mathematical reason. The order of operations is only a matter of notation to save some brackets.

Careful: a typical calculator does not have a different order of operations but none at all instead. So

5 - 4*3 on a calculator is actually (5-4)*3 while with our convention for algebra it is 5-(4*3).

Both assumptions are valid, the latter one is just the more common form.