How to get global screen coordinates of currently selected text via Accessibility APIs.

You can use the accessibility APIs for that. Make sure that the "Enable access for assistive devices" setting is checked (in System Preferences / Universal Access).

The following code snippet will determine the bounds (in screen coordinates) of the selected text in most applications. Unfortunately, it doesn't work in Mail and Safari, because they use private accessibility attributes. It's probably possible to get it to work there as well, but it requires more work and possibly private API calls.

AXUIElementRef systemWideElement = AXUIElementCreateSystemWide();
AXUIElementRef focussedElement = NULL;
AXError error = AXUIElementCopyAttributeValue(systemWideElement, kAXFocusedUIElementAttribute, (CFTypeRef *)&focussedElement);
if (error != kAXErrorSuccess) {
    NSLog(@"Could not get focussed element");
} else {
    AXValueRef selectedRangeValue = NULL;
    AXError getSelectedRangeError = AXUIElementCopyAttributeValue(focussedElement, kAXSelectedTextRangeAttribute, (CFTypeRef *)&selectedRangeValue);
    if (getSelectedRangeError == kAXErrorSuccess) {
        CFRange selectedRange;
        AXValueGetValue(selectedRangeValue, kAXValueCFRangeType, &selectedRange);
        AXValueRef selectionBoundsValue = NULL;
        AXError getSelectionBoundsError = AXUIElementCopyParameterizedAttributeValue(focussedElement, kAXBoundsForRangeParameterizedAttribute, selectedRangeValue, (CFTypeRef *)&selectionBoundsValue);
        CFRelease(selectedRangeValue);
        if (getSelectionBoundsError == kAXErrorSuccess) {
            CGRect selectionBounds;
            AXValueGetValue(selectionBoundsValue, kAXValueCGRectType, &selectionBounds);
            NSLog(@"Selection bounds: %@", NSStringFromRect(NSRectFromCGRect(selectionBounds)));
        } else {
            NSLog(@"Could not get bounds for selected range");
        }
        if (selectionBoundsValue != NULL) CFRelease(selectionBoundsValue);
    } else {
        NSLog(@"Could not get selected range");
    }
}
if (focussedElement != NULL) CFRelease(focussedElement);
CFRelease(systemWideElement);