Unfortunately Widget configurations are stored on each individual page they are used on. This is done in an XML format, making querying and updating them in bulk pretty hard.
Depending on what is stored (always same value? property name?, etc...) you could try to SQL-script it directly on the DB (don't do this on production, and always create a back-up!), or create a custom module that loops through them.
Using SQL script is also a good way for just identifying how many (and which) pages you need to update so you can do it manually.
Whit regards to the property being visible, each property should be able to made visible on the editing form. Can you share the settings? for that property? Does it have depending fields rule, what is the control used? Any Macro's on the visible setting?