Domain Mapping and WordPress address

Hi there!

I’m interested in using WordPress for my Business website. I currently own a domain with a website company, but would like to move away as it doesn’t offer the flexibility that i require.

The domain name i was hoping to set up to create the website is taken (Just a free WordPress version). My plan was to have my old domain mapped once the WordPress website was set up and ready to go. Id like to know if having a different name when i set up WordPress matters when i go to map my current domain later.

Which leads me to my second question… Is it actually possible for me to map the domain once iv’e set up the WordPress site? or would i have to create a new site after mapping?

Id also like to know if, when i do map the domain, i could make it example.com.au instead of example.com . This is very important for my business and if i cannot make it .com.au, i probably wont use WordPress.

Thanks in advance for your help.