Second question first: Yes, Windows Explorer does that. It shows 1 in the Track Number Column when it sees 01. All that matters is what is actually in the field, which you can see with mp3tag.
But I don’t understand why you’re having track order problems IF you ran the Auto-Numbering tool while they were in the right order. The Fuze should read the track numbers from the track number field.
Sometimes I get albums that didn’t make track numbers part of the filenames. So you get the songs alphabetically. A-song, B-song, C-song, D-song. If mp3tag has them listed alphabetically by filename, and you click the Auto-Numbering Wizard, A-song will become track 01, etc. mp3tag doesn’t read the existing track numbers in that case–it numbers the files in the order they are listed, top to bottom.
If you do have filenames like A-song, B-song, etc., but they are correctly numbered in the Track field, you can click the Track header to put them in track order. (Clicking any other header will also regroup the files.)
Another cute mp3tag feature: Convert. It can turn tags into filenames or filenames into tags, using the fields you specify by putting %track%-%artist%-%title% (etc.) between percent signs. My example would read the tags and rename the files 01-Artist-First Song Title. So if you had those alphabetical titles but with correct track numbers, you could use Convert Tag to Filename to make them 01-First Song, etc. And then they would list correctly in Windows Explorer.