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The intuitionists argue that mathematics precedes logic, whereas Hilbert and his followers (their position being Platonism today) argue as you do (mathematics has its roots in logic).

Both branches of mathematics disagree on basic logical principles (for a Platonist "A or Not(A)" is universally true, but for an Intuitionist it is provably false in some instances). This leads to simple properties such as trichotomy on the reals (given any number, it is <0, =0, >0) failing for intuitionism but being valued for the Platonist.

Godel's incompleteness essentially tells us we can never know which position is "the right one", as no system can prove it's own consistency (i.e no system can ensure itself will never lead to a false result).

Both are acknowledged as consistent systems with respect to one another within academic journals. It is very much a matter of philosophy as to which one is accepted as true.

Math is generally pure logic, but based upon arbitrary rules or unproven assumptions. In your example you assume the bag is empty and that it can hold at least 5 things.

For example, Euclidean geometry is based on the concepts of points, lines, and planes. All of those are assumed to exist as defined with no proof.

In a similar vein, philosophy generally defines a base assumption as true and works from there to form a framework.

If you put two things in a bag, and put three more things in there, there are now five things in the bag. I don't think there are other valid answers.

Math starts from that basic simple assumption. Then it can slowly dog-paddle into the weird end of the pool that blurs the line into philosophy or epistemology.

"If I have five things in the bag and I cut one in half, do I now have six things in the bag? What if cut one of them infinity times? Are there any exceptions in which two plus three may NOT equal five or fifteen times twenty may NOT equal three hundred? In either case, how do I prove it? Can I prove it in 24 dimensions?"

But yeah, even when exploring new math, it's still like a numerical story that's writing itself, as if the mathematician is taking dictation, as opposed to making stuff up, because then the math doesn't work.
If Newton and Leibnitz had not come up with calculus, somebody else would have, and sooner rather than later, conditions were ripe for a tool exactly like calculus to be useful. And it would have ended up looking exactly the same.

Is this ALWAYS the case? Is there any math that is purely creative? Can there be any math like this? Maybe the academic world is rife with this sort of thing, and I simply don't know about it.

That is indeed the case yes!