Is there a shortcut or way to use code, to do anything at all with the item currently highlighted by Command+F?
If you select text in Safari (your example) Accessibility Inspector will list among Attributes:    


"Passive" text . . AXRole: AXStaticText &nbsp; / &nbsp; AXValue: [text of a whole paragraph]
Link . . . . . . . . . . AXRole: AXStaticText &nbsp; / &nbsp; AXValue: [link name]


There's not very much functionality to detect there, is it?
Selected text in TextEdit on the other hand will display:


[Any text] . AXRole: AXTextArea &nbsp; / &nbsp; AXValue: [Whole text] &nbsp; / &nbsp; AXSelectedTextRange: [value]
[Link] . . . . AXRole: AXLink &nbsp; / &nbsp; AXRoleDescription: Hyperlink &nbsp; / &nbsp; AXURL: [its URL: http://...]


In Safari not too much can be done with highlighted text. Other programs -TextEdit!- that support accessibility more naturally, CAN be manipulated to a higher degree.   

If you look at the hierarchy of a quite common button ("edited" on this page) it will reveal itself as:   

AXStaticText ["edited"] / AXLink / AXGroup / AXWebArea / AXScrollArea / AXGroup / AXGroup / AXTabGroup / AXSplitGroup / AXWindow:AXStandardWindow / AXApplication   

Doesn't sound very inviting to me, searching for a simple button ...
If I had to do a search I'd rather look into a Safari-document's source, text or URL, all of which are accessible, see: Safari's sdef file.