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Daniil - good questions - different people may want to weigh in on this with ideas. My own suggestions are that yes the pawns from the queenless side can promote into a queen and that the pawn starting on the queen square can advance one two or three squares for its move (that en passant could be exercised against it on the double or triple space move), but that after that initial move, the double (or triple) option is taken away (which might be hard to remember if this pawn opts to move just one square for its first move). I'm happy to see that this game has been fully played out several times now since I posted it here. I'll be interested in reviewing the games...

What does anyone think? Is one side superior? Or are they evenly matched?

Perhaps in Pawnless vs. Pawnful original Good array, one side has forced win.  That is what Muller demonstrates for Maharaja and the Sepoys: http://www.chessvariants.org/index/displaycomment.php?commentid=24978.  
M&S is similar in being Chess Diferent Armies like P v. Pawnful.  My conjecture elsewhere is that Maharaja should win on 10x10 against same 16 aligned Grand Chess-style, because the Muller conditions cannot be met on the larger board.

In the several games played of Pawnless vs. Pawnful, was there any Queen promotion by the Queenless side? The pieces are always going to win, but it still has to be proved.

http://www.chessvariants.org/unequal.dir/dunsany.html, 32 Pawns should win not only because of 32-31, also because of the board advantage.  In other words, again expand it to 10x10 100 squares and the pieces+8 probably win.