Hamza is a letter (basically it is what the letter alif is supposed to be, the long vowel usually written as alif is not a letter itself). But for historical reason it has a very weird set of orthographic rules where is can or can not be seated on other letters depending on the context (and unlike other Arabic letters, the context here involves the vowels).

It isn't. It can be combined with other letters, mostly with Alef, but I don't know of any case of it being used independently.

The reason why I would say I think it is a letter, is because it can be a letter in a root. Like in قرأ, the letter Alif is not considered the third letter in the verb, hamza is. I had to look it up, the grammatical term for it is الفعل المهموز, and all of the examples I can find have it placed on the alif in the root form, but it changes depending on how you conjugate it.

Also, what about the word for "thing" شيء it's not placed on the yaa, but written by itself. Or the plural form, things: أشياء has two hamzas in it, only one sits atop an alif.

So that's a thing.

It actually can. An example would water: ماء