Yeah, the O(1) refers to the size of the EXTRA memory you need, aside from the input. Defining it any other way would be strange. So, for instance, binary search over a sorted list needs O(log n) of memory (for the same reason as the prime algorithm, the pointer in the array is size O(log n) of the size of the array), even though the input is O(n).
Yup. Note that the method of specifically being fed the input sequentially, one by one, means that, for example, "Strings of odd length whose middle character is 'X'" does not comprise a regular language, even though one might naively reason this to be trivial to detect with O(1) memory ("Just go look at the middle character!").
