Please submit your issues here on our bug tracker on github.
Can you provide more reproduction steps? When we try to reproduce this, the dropdown does go away.
You would need to put the widget in the Categories Siderbar section. If you don't have this section, your theme needs to be updated. see here for needed changes.
@Jam can't replicate it... using master
Downvotes weren't disabled I think the forum was pushed with the wrong vanilla version, it's borked the tags page as well lol
Ah... then if mobile is definitely getting a different composer, then a resized desktop composer I'd accept the compromise of: < 768px? hide preview box altogether