1/ When which file is created? If there's a cover, then there's a cover... o_O
2/ Nope. Calibre automatically takes the cover that's currently set in its [Calibre's] metadata for the book when you do a MOBI/AZW3 conversion (or a Send To Device), and embed (or replace) it in the file [and update the metadata pointers if need be]. No idea how other readers handle mobi files, but this is pretty basic stuff, so I'd wager that everyone does this properly and honor the file's metadata/content.
3/ Not exactly. The embedded thumbnails is used when there's no ASIN set, or you're using JBPatch. The external (or sideloaded) thumbnail is used otherwise.
The thumbnails are in the system/thumbnails folder.
Cover mode always takes precedence over everything else.
In cover mode, the behavior at boot is to use the last screensaver used. If it's not a fresh install, then, yes, it will use the last thing used (either the last screensaver, the overlay from the 'last' mode, or the last cover shown) until a book is opened.