We can't tell you why the file dates changed except that a Move wouldn't do so (or shouldn't - depends what you Moved them with). But moving files to a new location may have triggered other indexing software (and/or media applications) to process them, and that processing can sometimes change their properties
If you cannot select Copy to Source, this means your selection includes files that are a mixture of states/locations, and have included files that are on Source only. As you cannot 'copy to Source' files that are only on Source anyway, that context-menu option is suppressed. But you can 'Mirror to Source' a mixture of files where
some of them are Destination-only > > > would be copied to Source
some of them are Source-only > > > > > would be deleted from Source
because Mirror implies potentially more than one action (Copy some files, Delete others)
'Use details from [side]' means copy only the attributes of the file on that side (to the same-name file on the other side), and not the entire file-contents (as per the Help...)
If you cannot select Copy to Source, this means your selection includes files that are a mixture of states/locations, and have included files that are on Source only. As you cannot 'copy to Source' files that are only on Source anyway, that context-menu option is suppressed. But you can 'Mirror to Source' a mixture of files where
some of them are Destination-only > > > would be copied to Source
some of them are Source-only > > > > > would be deleted from Source
because Mirror implies potentially more than one action (Copy some files, Delete others)
'Use details from [side]' means copy only the attributes of the file on that side (to the same-name file on the other side), and not the entire file-contents (as per the Help...)