When selecting a previously entered search string in "Search and Replace" dialog with Alt+Up, it should be remembered and restored whether the search string is a regular expression and automatically set/reset "Regular expressions" checkbox.
One almost never does regex search with non-regex string or non-regex search with a regex string.
Probably also should be stored and restored "Ignore case", "Whole word". It is disputable.
Maybe also the replacement which was used? So "scrolling" through the previos search strings would also scroll throught the replacement strings at the same time.
With such unconditional synchronization flexibility is reduced, switching back from replace string after its modification to search one is no longer possible. Isn't it?