Sounds like a lot of problems.
Moving the database/library should copy over only those folders that are actual library entries.
saving to disk will create an opf file that will associate the metadata with the file and you can save to disk so all the files are in one directory.
The underscore in the folder name seems to happen when you have a comma(or other illegal filename character) in the author name and for me the extra folder is removed when the name is corrected. It used to be easier to spot in the author list when it was displayed alphabetically by first name. Now asmall difference in name can be several entries away. (all the Smiths would be listed in order and then a Smith, would follow)
Sorting generally works for me but is a little odd once in a while. For me it is confusing to not be sorting on what you see but I am sure there are good reasons for it.
typing the author name into the search bar is often an easier way to go. type in name, then sort on author should help you narrow it down.
Thanks for posting as I did not realize that I had the odd underscore in there myself
Helen