The sqlite is only storage server-side. Otherwise you would have bazillion of small files in your filesystem and they mostly do not handle that well. (Speaking of experience of having 800k tiles in a single NTFS directory - that was not pretty).
When serving the tiles over http, the only tiles themselves are served, not the entire sqlite database.
This was actually pioneered in the era of raster tiles - pngs/jpgs in the sqlite db. Vector tiles just "inherited" that.
No; however there are not many vector tile formats publicly defined. MVT is good at what it does.
The generic vector files used in GIS (shapefiles, DGN, KML, GML etc) are not a good fit for tiles - they describe a specific classes of features and do not optimize for tiling (i.e. you must always parse the entire files without regards what extents you are really interested in) - which is fine for generic GIS usage, but not for end-user vector maps.
When serving the tiles over http, the only tiles themselves are served, not the entire sqlite database.
This was actually pioneered in the era of raster tiles - pngs/jpgs in the sqlite db. Vector tiles just "inherited" that.