1.Is it incorrect to add “si” and say “C’est fou comme ma boîte de réception se remplit si rapidement !”?

You can say that, it is correct but I would not use it this way. I would rather say “Ma boîte de réception se remplit si rapidement” and remove the “C’est fou” because the sentense becomes heavy.

2.In order to express the idea of “How fast … !”, does the word “rapidement” always have to be placed at the end? Is there a way to move “rapidement” closer to the top?

The placement of “rapidement” is very adequate.

You can also use “vite” instead. As a native speaker I would say “C’est fou comme ma boîte de réception se remplit vite”

3.You cannot substitute “comment” for the exclamative sentence “How … !”, correct?

ah ah, if you would say “Comment ma boite de réception se remplit rapidemment !” would not be proper french technically speaking. I would say that to collegues but not to my boss for instance.

4.Is it possible not to use “comme” in this sentence?

You can remove “comme” but then you would have to replace it with “,” like so :
“C’est fou, ma boîte de réception se remplit rapidement” or make two different sentenses “C’est fou ! Ma boîte de réception se remplit rapidement” or say “C’est fou comme ma boîte de réception se remplit rapidement”.

1. Is it incorrect to add “si” and say “C’est fou comme ma boîte de réception se remplit si rapidement !”?

IMHO, if not incorrect it is not idiomatic. The si here would add nothing and native french speaking people would not use it.

2. In order to express the idea of “How fast … !”, does the word “rapidement” always have to be placed at the end? Is there a way to move “rapidement” closer to the top?*

If you use the adverb, it has to be placed there. You would have to use the noun to put it closer to the beginning: C’est fou la vitesse à laquelle se remplit la boîte de réception. I used vitesse instead of rapidité, because the latter is less common in spoken language, but more on written/formal one.

3. You cannot
   substitute “comment” for the exclamative sentence “How … !”, correct?*

Yes. How … ! is an idiomatic english expression that cannot be translated literally. The almost litteral translation would be Comme une boîte de réception se remplit vite ! but it would be rather written language. The C’est fou here could be replaced by a (familiar or popular) interjection: Bon sang (ou Putain) qu’une boîte de réception se remplit vite !. IMHO it depends on the expected language level.

4. Is it possible not to use “comme” in this sentence?*

The expression proposed for the second question does not.