I don't quite understand why the new Senate does not come into effect until 1 July, operating with the previous election's results, when the new House of Representatives sits almost immediately after the election. What is the reason for this inconsistency and time delay?
I can tell you where it comes from, but not why it's the way it is.
The Australian Consitution, Chapter I, Part II, Section 13 says:
For the purpose of this section the term of service of a senator shall be taken to begin on the first day of July following the day of his election, except in the cases of the first election and of the election next after any dissolution of the Senate, when it shall be taken to begin on the first day of July preceding the day of his election.
There is no such provision in the equivalent Part addressing the The House of Representatives.
Senators also serve a 6 year term, roughly half of which expire every 3 years. The House must be reelected every 3 years, although it can happen more frequently. It just so happens* that currently the term of the House is such that it's convenient to elect the next batch of Senators, but that is not a requirement.
* And by "just so happens", I mean "it was arranged such"