Having grown up on the Northern Beaches of Sydney (bounded by Manly in the South, Palm Beach in the North and Belrose/Forestville in the west is a GREAT place for kids to grow up. That's where we are headed with 19mth old Son and baby on the way come March.
If your funds don;t stretch to the rents there, the Hills District is a good place to start - Baulkham Hills, Carlingford, Thornleigh etc etc.
If renting look at
Real Estate, Property, Land and Homes for Sale, lease and rent - realestate.com.au and search by 'map' and you can then narrow it down by area eg Northern Beaches, Hills District etc.
I really don;t think there is a limit to getting a mortgage, or credit cards for that matter. If you are really worried, you could set up an HSBC account in the UK and then transfer your funds to HSBC in Australia (they have limited branches). I got a mortgage in Oz (I am Aussie, DH is a Brit) despite not having been a resident for 5 years, so you should be fine as long as you satisfy them that you have regular employment etc.
Good luck and congrats on the Visa!!