in a polynomail function, does the leading coeeficient do anything else than tell you the direction of the graph (i.e. whether it's inverted or not)?

Answer 1

Yes, it tells us how steep the graph would be.

f(x) = 2x²+x+3 would be steeper than
f(x) = x²+x+3
In addition, the zeroes would change in position.