Hi @evert823, this is a very good summary and I especially like all the diagrams to explain various situations.
My only suggestion is that this topic should start with the key rules, and then other secondary details (such as "no en passant") should be secondary. This means the total rule-set would read as follows:
Bulldog (D): Moves and attacks the same as a pawn, except pieces FROM ITS OWN ARMY ONLY can pass over it in any direction. If a bulldog reaches the 8th rank it can immediately move to any square in the first four ranks (but cannot capture a piece during this move). This move can only be completed during the same move it reaches the 8th rank. (If this option is not taken, the right to do so later is forfeited). The bulldog does not promote to other pieces as pawns do. Bulldogs cannot take en passant, and they cannot be taken en passant by pawns.
Note: The bulldog is a pawn-like piece. In the initial setup the bulldogs should start on the second rank in place of two or more pawns.
Bulldog:
Inventor: @vickalan
Notation: D
The Bulldog moves and captures like a pawn.
Pieces FROM ITS OWN ARMY ONLY can pass over it in any direction.
In this diagram, the Black Rook attacks the White Rook. The White Bishop attacks the Black Rook.
This can also be put in the same terminology as the Witch: The Bulldog is transparent to allied pieces (but does not make adjacent pieces transparent).
If a Bulldog reaches the 8th rank, it can immediately move to any square in the first four ranks (but cannot capture a piece during this move). This move can only be completed during the same move it reaches the 8th rank. If this option is not taken, the right to do so later is forfeited.
Notation: Dc8-h3+ (+ indicating as usual that it gives check)
Further notes
White plays d4. Black captures 2 pieces in one move with en passant exd3 (e.p.)