Depends on your policy.
Your mortgage company will insist on building insurance obviously. Some of these policies have varying degrees of contents covered as part of the deal. Some don't.
So would the shed be insured as its contents but not part of the house 
Depends on the policy. Usually not. As the base level insurance of "building insurance" only covers what's part of the mortgag loan. I doubt your paying a mortgage that includes a shed. Which is the typical scenario.
You need to check your policy but I'd be suprised if shed contents are covered as minimum level of cover, unless you specified it.
But then, some class an out building or coal bunker as a shed, and if its a lean too type out building attached to the house, the "shed" might be classed as part of the building.
Depends on the type of shed. Some sheds come under vehicle insurance.

As a VERY general rule, Building insurance usually covers what does not fall out if you turn the dwelling upside down and shake it. What does fall out, you need contents insurance for.