Files vs Playlists

From what I can interpret from your description, the 'Files' menu system is representing the file system of your Tab, the actual music files (probably mp3, aac, or possibly ogg) inside the folders your created.
The 'Music' menu system you refer to is probably what you're seeing through whichever media player app you're using. If so that's a matter of the media player showing what it's designed to show, in this case just the music files on your Tab, not necessarily their specific location where they're being stored.
As for your playlists, keep in mind a playlist file is just a text file with the name and location of a collection of media files. You can open one up with a text editor and you'll see something similar to this:
/home/Audio/ChicagoBluesFest1990_JohnLeeHooker.ogg
/home/Audio/ChicagoBluesFest1990_RuthBrown.ogg
along with a lot of extraneous text, but the point being the playlist file itself is just simple directory, telling a music player the name and location of a music file. By transferring your music files from one device to another, the playlists are for the most part 'broken', the info they contain pertain to your music collection but the actual file path, as in where they're located, is now completely different. The easiest thing to do is just create new playlists so the information inside them is current. Another reason to make new playlists are issues tied to how different operating systems rely on different filesystems. Some things are just annoying, like how Unix based systems (Android runs off a Linux kernel) use forward slash / but Microsoft has opted to have Windows default to using backward slash \. There also more fundamental problems as far as how different operating systems (and different media players) interpret file metadata -- possibly part of the reason why some of your album/genre stuff didn't transfer consistently.

 

You might want to look into a mp3 editing app to help set up your Tab:
https://play.google.com/store/apps/details?id=com.ADnet
https://play.google.com/store/apps/details?id=mobi.infolife.itag&hl=en

As for using your Bluetooth speaker with your iPod, some iPods support Bluetooth, some don't, it depends on what model you have. Also, some Bluetooth speakers only have the wireless Bluetooth input, some have both Bluetooth along with a port to plug in a cable.

 

The iPod would have been using id3 tags to look at your music. Those wont show up on file names. The music player on the Tab should also use that same data, as well as music players on a PC. These data are separate from file names. You could right click the file and check in properties to see and edit these tags.

 

I have to disagree with that. As I was alluding to earlier about metadata, it's not a universal, OS neutral issue. Usually only small subset of metadata can actually be altered through 'Properties' and this can vary widely especially when you're working the same files that have been transferred between two or more operating systems.