"So, you have 3 small apples, and 1 large apple.
How many apples do you have?"

3 Small apples and 1 large apple. This is how you broke up apples by size.

"Does 3 apples x 3 apples make any sense when there are only 8 apples in existence?"

Of course not. If there are only 8 apples than there is no way to verify and confirm the 9th.

Multiplication only multiply's 3pens x 3pens, which equals 9. Thus there must be 3 Pens to multiply 3pens. So one of the integers does not have 3 Pens and in fact there are possible 2Pens in one of the sets and is why one must verify and confirm the symbols represent a something.

"And since we can never count to infinity, does infinity exist?"

Is a hypothesis that can never be proved and as such not math.

"And if it doesn't, what happens to calculus?"

When ever one cannot count, verifying and confirming the integers  one does not have math. It really is that simple. 

"Further, 1+1=2 is an assumption. It is not proven. It may not be provable.
2+2=4 was proven recently, as long as you assume 1+1=2 and that integers exist."

No 1 + 1 = 2 is not an assumption unless you don't verify and confirm your symbols represent what is there. So without verification and confirmation one is assuming. With verification and confirmation one is determining based on the rules of math the truth of the condition. 

"I can see what you are trying to get at, but reality is not as solid and set as one would like. There isn't a line where reality ends and imagination begins.

Or, in other words, half of our brains are on the other side of that line."

I totally agree with the entire statement and add the following.  Math is about the somethings that are solid and countable. When ever counting is not possible what ever you are doing it is not math.

The confusion is a direct result of people not being required to verify and confirm there integers for if on a regular bases you have to count 3 Pens and than X 3 Pens you would comprehend that the first integers must represent a something just like the second integer must represent a something.   When a integer is not representing a something it's value is 0 no matter what the symbol implies.