potentially yes.. Visa does not generally change tax liability (LTR excepted), source of income time incountry and DTA does.
There is also the issue of liability v assessability, and the declarations that may be needed to ensure compliance (at this point even professionals disagree).
No one can answer this.. Immigration HQ have been 100% silent on availability and requirements for extensions.
Various regional offices have made a range of comments, which are best categorised as guesses and a cultural inability to admit 'I dont know' at this point.