1) Tableau Pricing depends on what kind of infrastructure you're planning on setting up, number of users, licensing model itself. Here's a link for more information on Tableau Pricing options
2) You need to check with Tableau Account Manager of that company.
3) Tableau workbooks can be shared in 4 ways - Server, Online, Public, Reader. You can additionally host the workbook on web or SharePoint. I'm however not aware of any other hosting platforms.
Hope this information helps.
The best advice would be to ring Tableau and talk to them. These forums are largely run by the community and not Tableau employees at all.
That said ....
1) You need Desktop to create dashboards, you need Server or Online to distribute (well, that's the cheapest options) Detailed pricing is available on the Tableau website, but largely depends on exactly how you want to deploy
2) I have clients that allow external companies to access their workbooks. Whether you pass this charge on to the client or not is your decision. Either way, you need a license for every user that logs in.
3) "third party cloud service" - sounds like you're more thinking of Tableau Online. Tableau Public means that ANYONE can see and use your workbook, you probably don't want that unless you're using 100% publicly available data.