How can I create network with different domains?

I have heard similar questions posed but not exactly what I am looking for.

Is it possible to create a multisite with different domains and sub.doamins?

For example, can I create a network that allows me to create build:

SiteName1.com is a BuddyPress video community.

DomainName2.net is an online shop that sells products.

TheDomain3.co is a web design hosting company that creates and manages other websites. Users must be able to create subsites of their own like (username.TheDomain3.co) Also if they upgrate their site then username.com would replace username.TheDomain3.co

Also website4.uk would be a central affiliation centre that would like all the others together and offer people referral commission on sales and views.

I know all this is possible through MultiSite: but my question is this: Can I build this network so that each site has its own unique domain and if the user logs into DomainName2.net and adds a product to his/her basket I would need the user to still be logged in when they visit website4.uk and see the product in their basket.

I would like to make sure basket, history, activity, accounts/profiles/channels are synced throughout the website.

A perfect example of what I want my site to be like is ThemeForest.net if you add a theme to your basket and then visit codecanyon.net you can see the shared basket of the multi sites.