I took a look at both the 'Search' and 'Replace' dialogs and both  already have wrap-around present as the default: '[x] Wrap around'. Sorry for not checking earlier.

Having the find window stay open will make it much easier to visually check all the options.* 

The Edit menu has 3 find entries
Find ...        Cntl+F
Find again      Cntl+G
Find selection  Cntl+H

I think these menu items should lead to a 'Find' dialog, not a 'Search' dialog. I find the name switch confusing. Both 'find' and 'find_replace' search the text (and both dialogs use SearchDialogBase). That would be a trivial change to add to the patch if everyone agrees. Otherwise it can be a separate issue.

Re-considering whether we still need those 3 entries after this patch would be a separate follow-up issue.

* For automated tests, is the Idle code currently organized in such a way that we can code the following without actually running Idle or showing anything on the screen: create an edit widget or window with some text, create an associated dialog box, set and unset text and options in the box, call an 'action' function, and then check in the edit window that the 'correct' text is highlighted and that the cursor is where it 'should' be? (This of course requires exact specification of 'correct' and 'should'.)