Hello,
1) I note there is a known issue with the search bar not staying at the selected location when the CPT is changed. I understand this will be corrected on the next release #468639. Any ideas when – roughly?
Not exactly sure about next release, probably within next week. I have updated patch on your site to fix this issue.
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2) In the “Search for …(Optional)” box, when a category is selected from the drop down box, the page automatically refreshes to make the search. Should this not wait until the search button has been pressed as is the case with the contents in the other two boxes?
This is default behavior in search. In you select category/listing from autocomplete results it will take you to category/listing page. If you want search within particular category then you can add category field to search form from CPT Settings > Search > Category.
Let us know.
Thanks,
Kiran