Good morning Gnome,
These are good and valid questions.
1/ Yes, an L1 can maintain his own (24 xxxx) RA Aus aircraft, for his own use, so long as it's not being used for commercial training.
2/ A factory built aircraft can be used for training, even after it's been maintained by an L1, BUT, it must be inspected, maintained and signed for by an L2 or above BEFORE, and then WHILE, it's used as a commercial trainer.
Questions 3 & 4 tie in and are a little more complex, so I'll answer them together:
In the case of any aircraft, syndicated or not, one person is nominated as being responsable for the aircraft's maintenance. That doesn't mean he has to do the maintenance, but it does mean he is responsable for ensuring the aircraft is maintained in accordance with the Maintenance Manual (MM), Service Letters (SLs), Service Bulletins (SBs), Airworthiness Directives (ADs) etc. by competent and qualified people. Now, depending on the syndicate members, that may be a big ask, or not, but it is required, and therefore finding someone to accept that responsability may be a possible downside in a syndicate owned aircraft.
Hullo, I've just realized while sitting here writing this, that I don't have the reference material in front of me to verify and explain the maintenance requirement complexities of training in a syndicate owned aircraft (it's out in the hangar) so I'll have to get back to you... To be continued...
Edit: Before I go, it may be quicker to pick the phone up and call Adam Finn in Canberra. He'll probably be able to give you a definitive answer quicker, and far more authoritative, than any interpretive answer of mine.