We\'re using an API from a provider; in their next version, they would provide an OpenId server but mostly the users are using their desktop application. An other website we\'re
OpenID and desktop apps are totally completely and altogether incompatible. You cannot secure a desktop app using OpenID.
Look at OAuth. It can work with or without OpenID, but OAuth is all about authorizing apps including desktop apps to access network resources.
I think the answer is "it depends." For the close-and-reopened browser question, it depends on whether your sites use session cookies or persistent cookies to authenticate requests.
But as for the authenticate-with-desktop-client-and-then-open-a-browser, you'd have to ask the provider for more details. I can imagine ways to do it that would have either outcome.