How can we open a link in private browsing mode
That's up to the user. You can't force a user to open the browser in private browsing mode.
HTML a hyperlink target inPrivate / incognito window
We can't force the visitor to view the page in an incognito/private window. Browsers provide no API that would make that possible outside of an extension.
Is it possible to create a link or button that will launch the link in incognito / private mode?
It is not possible on a webpage but it is with an extension as there are many privacy concerns if a webpage can open tabs in incognito / private mode
Is it possible to have a link that always opens a new tab in 'normal browsing mode', even if clicked from incognito?
This would be possible in a browser extension using, for example, chrome. windows.create which can open a window in an incognito or normal window. It may not be possible in all browsers, but TamperMonkey, GreaseMonkey & friends would be worth checking out for that possibility.
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