How to backup the wordpress.com hosted blog

There is NO way that I know of to do a complete backup (defined as an off-line copy that you can restore from).

You can export your post and comments via Settings > Basic > Export Blog - and this will let you re-import those posts and comments into the same or another (Blogger) blog.

You can save your template via Design > Edit HTML > Download full template - and this will let you re-import the template as it was.

You can make an off-line copy using something like HHTrack: This is useful if you need to show people you blog off-line - but you cannot restore from it. Also, there are some components (Google Maps for certain, I'm not sure about others) that the copy will have the links to, but not the component (meaning it won't work off-line).

I don't know any way to backup the contents/settings of your gadgets (your blog-template knows where the widgets go and what type they are, but does not appear to know what settings they have).

Custom domains are a paid upgrade.

The Domain Mapping Upgrade allows you to use a custom domain name such as example.com instead of the standard WordPress.com domain name – example.wordpress.com – for your blog. This is a paid upgrade.

Wordpress.com has a tutorial on doing this once you've bought the upgrade.