Site set up with mutiple sub domains I will build a group site in craft which will have the main domain and multiple sub domains.
e.g. 
domain.com
property.domain.com
connect.domain.com

The news feed on each sub domain will be different but it will all display on the main domain news feed. The site build etc for each will be similar but content will be different.

What is the best way to set this up in Craft?

Is the following something which can be done.

Use Multi Environment Config to set the SiteURL for each domain.
Point all the domain/sub domains to the 1 public folder.

Within my main _layout.html template file detect what the siteurl is and based on that include different header and footer files for each sub domain?

The same news template files can then be used. 
And just use a different structure for the nav on each sub domain.

Is that workable and the best way to do it or should I be having different root folders for each sub domain?
The approach you described is interesting and should actually work. Separate root folders would allow you to define the template folders individually (→ see Ben's answer on how to set this up), but it looks like that's not what you need and your subdomains are more part of one and the same site.

Even with multiple root and template folders you could still include shared template partials, but sharing CSS, JS and assets would be troublesome. But, then again, more easy to have these individually for each subdomain.