A viable prefix is a legitimate prefix that could lead to a handle.
Since handle will always be found on top of the stack, all the viable prefixes of it can appear below it in the stack. If handle is not found yet, the stack would contain just viable prefixes.
I is true.
Viable prefix can't extend past the handle. That's why it's a prefix!
II is true.
The set of all viable prefixes of a language is a Regular Language, so it can be recognised by a DFA.
III is true.