Graeme:
(1) & (2) Noted
(3) And the 1993 EA replicated the 1956 wording etc.
(4) No. But you will be aware that natural justice rights are contextual. It may seem to defy logic, but instinctively there is something different in character about ousting someone from democratic office and taking away their liberty, if the conviction was unsafe. The courts can’t restore an MP to office. However, they can order they be freed (albeit can give them back the days of liberty lost).

PS Happy for the wager. Bottle of pinot noir?
PPS Note to self: Must re-read Awatere-Huata before making any more wagers. The role of the Speaker and the effect of parliamentary privilege may be crucial.

Yeah. I'm not convinced.
(1) There's wiggle-room there in the manner in which the conviction is entered.
(2) There remains an interpretative question, perhaps for the Speaker, about whether "is convicted" is a snap-shot or means "is convicted" following exhausted appeals.
(3) While the LG Act has different language, it was drafted in a different era and isn't determinative.
(4) Section 27(1) NZ Bill of Rights (natural justice, read together with s 25) may suggest an interpretation that allows the exhaustion of an appeal before removal from office.
On the other hand, there is something about certainty and swiftness, even a successful appeal might demonstrate that to be manifestly unfair.

Graeme: Do you have authority or commentary addressing the point automatic effect of conviction - regardless of appeal - point? I've heard that said about s 55 before, but find it odd and suspect the courts prob work to avoid an appeal being nugatory?