Secondly prospective theorems are absolutely falsifiable. Since a theorem is a statement that has been proven to be true yes they are unfalsifiable by definition - they have already passed that test. That doesn't really generalise to any sort of meaningful statement about the falsifiability of maths. Saying theorems are unfalsifiable is equivalent to saying "True statements can't be proven false". Well, yes.[1]
ie If I say Sean Hunter's theorem is that if you take a triangle with arbitrary sides a b c and angle opposite a of theta that
a^2 = b^2 + c^2 -42 b c cos theta
that statement is absolutely falsifiable (and false), which you can establish with basic geometry and trig[2]. When you demonstrate it not to be true it is not a theorem, so I was wrong to call it that. That is a demonstration of how maths is falsifiable.
[1] Even so it's often possible to make progress via proof by contradiction - showing that if this theorem were not true something else which we know to be true would be false. But in most of my maths books proving all of the theorems is the norm, so they are for sure falsifiable while you are trying to establish whether or not they are theorems.
[2] Drop an altitude from one of the angles at b and c and then use pythagoras and a bunch of cancelling. You will prove that a^2 = b^2 + c^2 -2bc cos theta of course. My statement is only true if a is the hypotenuse of a right triangle meaning cos theta is zero and my incorrect coefficient doesn't matter.