If you upload files into Piwigo, they end up under the uploads folder with the YYYY\MM\DD folder organization and random filename. The various re-sizes, thumbnails, etc. are stored under the corresponding YYYY\MM\DD folder under _data\i\upload as you saw. You can then organize the photos into any number of "virtual" albums.

Using FTP synchronization, you define the folder structure under the galleries folder (physical albums). This is what I do as I like to control the folder names and hierarchy. To automate some of the manual stuff (like generating all the thumbnails), check out @rw22mhhs's new plugin, AlbumPilot:

https://piwigo.org/forum/viewtopic.php?id=34301

Yes, except for one thing to consider: if you physically move a photo from one album into another, and then run the sync in Piwigo to pick up the change, you'll lose all the metadata in Piwigo for that photo. If your details (like tags, name, description, etc.) are embedded in the file (IPTC/EXIF, etc.) then those will be re-loaded when the photo is "added" in its new location, but if not all of the metadata in Piwigo will then be empty.

I wrote a plugin that allows you to move photos from one physical folder to another, preserving all of that metadata in the Piwigo databse, [extension by windracer] Physical Photo Move.