I can't speak to the title issue but know many here can...
As for when you can register it into your own name, as I said, I tried and was told by ICBC that the vehicle must first pass the mandatory safety test, and only at that point can it be registered into my name... not a moment before.
If you wish to insure it while you're working on it in case it is stolen or damaged, get storage insurance... you can get this without being the registered owner and it usually costs under $30 per year...
~Trev
~Trev