I suppose an Australian would be of great help here!
Question 1: My name is Stephen (lemme know if you need last names too) and I'm from Sydney, Australia
Question 2: I would say there are two favourite sports in Australia that are equally popular. One is called National Rugby League (or NRL) and that is kind of like the international rugby you may have heard of from places such as Britain. This is played mostly along the East Coast of Australia. The other one is called Australian Rules Football (AFL) and that is a uniquely Australian game, which requires people to run around a circular field and kick balls between the posts. AFL is mostly played on the West Coast and in the Southeast, although it is picking up in popularity on the East Coast too.
Question 3: It is a bit difficult to include people with physical difficulties in the game (eg unable to walk) since the game is played on a large open field and usually requires a bit of physical contact (especially in NRL, where physical contact means tackling the opponent!). However there are many other sports that are popular in Australia that are more disability-friendly, such as basketball and swimming. On the other side, major sports teams in Australia play a huge role in Australia working with kids with disabilities etc and this usually involves playing a bit of sport with them as well. As for people with mental illnesses for example, they don't really play on a national scale for the major teams but generally there are smaller teams that they can play with if interested, so long as their mental illness would not lead to them causing themselves or others harm.
Question 4: Special rules or modifications? Well I suppose if a person in a wheelchair for example wants to play basketball, they would be placed in a team of other people in wheelchairs. Just a side note here, it was part of our school curriculum here that in Physical Education class we had to learn about disabled people in sports and as part of that we all had to play a wheelchair basketball game. Dunno if this helps, just thought I'd put it there in case.
I hope I answered your questions alright. If you need any more info please message me.