Can We Use a Different Variable (Such as Long or something similar) In place of Integer to get a larger Range?
Sure, any variable type that you can use in C#, you can use in Unity.
Whilst you can use a variety of data types, Unity’s Random.Range() accepts either int or float, depending which you choose to use, pay close attention to the documentation regarding inclusive and exclusive values.
You could also use C#'s Random class.
I have also included below a link to a rather well defined RandomProvider class which provides support for a variety of other data types.
It is worth reading up on some more of the intricacies regarding “random”, as your results will invariably never be truly “random”, just seemingly, but most likely sufficient enough. 
See also;
Unity - Script API: Random.Range
MSDN : Random Class
Cambria Research : RandomProvider Class
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