Paul, I assume that you’ve checked your Strata Plan and confirmed that the fence that you’re talking about is on the boundary of your lot and the one adjoining, and therefore that’s it’s a dividing fence and not a boundary fence. It sounds that way to me, but for your info. the former is a matter between you and your neighbour and the latter is an Owners Corporation (O/C) matter.
To answer your questions on the assumption that they relate to a dividing fence:
1) NO – but it’s the neighbourly thing to do.
2) On the basis of your intention as outlined at your Q3, NO, but again that would be the the neighbourly approach.
3) YES and totally appropriate provided 1 & 2 (above) are followed.
4) I can’t comment on peoples’ legal rights except where relevant Legislation applies ( e.g. The Dividing Fences Act in NSW), but as what your proposing on your side of the boundary could be an “addition to the Common Property” depending upon the classification of the land at the precise location, and/or something affecting “the appearance of the lot”, you would need the written prior consent of your O/C whether or not it wishes to become involved.