Can we apply searches on the server?
Not currently, but this will be possible in the Adaptable Blotter Server Edition which is scheduled to be released in 2018.
Can we ship the application to users with existing searches?
Yes you can create as many 'predefined' Adaptable Blotter Objects as you require including Advanced Searches.
Does Advanced Search update if the underlying data changes?
Yes it does. For example if you make an edit which means that the row in question is now longer part of the search results, the row will disappear as soon as you commit the edit.