In answer to your questions:
1. Perth is very cycle friendly. I have never been given a hard time while cycling.
2. Many people commute by bike. It's not difficult to do all year round because of the mild weather. My one-way commute is 17km and it usually takes about 40 minutes.
3. I'm not into off-road cycling but I believe there are many tracks in the hills as well as the 900km Munda Biddi trail.
4. I am not sure, but considering how popular cycling is in Perth you shouldn't have much difficulty in finding a good club.
5. There are many good shops, and probably a few not so good. Xtreme cycles in the city seem to be a good place for mountain bikes, and Glen Parker cycles in Nedlands seem to have the best ie exotic road bikes.

Are you transferring with your work and if so are you in the oil & gas or mining industries?

Cheers.