tl;dr: do M-x customize-variable for the variable help-at-pt-display-when-idle and set its value to t. (Or "always", which is what the Customize buffer says.)
What was initially confusing here is that there's the mouse pointer, and also the cursor (which here on emacs.sx we call point). As mentioned above, org-mode by default sets the :help-echo property ...
It appears that the function to call to browse a link in an org-mode file is org-open-at-point which is bound to C-c C-o.
Regarding the follow-up question about opening an org-mode link in a secondary browser, using another function seems to work:
(defun org-open-at-point-with-firefox ()