The above details are based upon the Superchess version, on the description in a booklet that I have at home, and on one verbal clarification during a Superchess tournament regarding this point: "If the opponent has promoted a pawn, the Joker inherits movement and normal capture abilities from the piece it promoted to".
The Joker explained

Is Jc1 legal?
Valid point. After Jc1 the Black Joker acts as Rook, but it is not check.

Would it "promote" (that is, act like a queen, knight, rook, or bishop, whatever choice you choose) instantly if someone moved a pawn and you have a Joker sitting on the 8th rank?

Also, in the Joker Tournament, could you change the letter J on the Joker to a piece symbol depending on what the Joker acts like? And if it's acting like a pawn and promotes, I would like the symbol to change to whatever piece you want to promote to?

And why doesn't the Joker inherit special abilities, but lose them if the opponent moves a different piece on his next move?

What if a Joker is on the 1st rank and someone moved a pawn?
It can move one square to the 2nd rank.
Why don't you just join the tournament?

Would it "promote" (that is, act like a queen, knight, rook, or bishop, whatever choice you choose) instantly if someone moved a pawn and you have a Joker sitting on the 8th rank?
No. As already said, a Joker does not promote. The consequence of sitting on the 8th rank and imitating a pawn is that the Joker can't move.
Also, in the Joker Tournament, could you change the letter J on the Joker to a piece symbol depending on what the Joker acts like? And if it's acting like a pawn and promotes, I would like the symbol to change to whatever piece you want to promote to?
No. I don't yet have a nice graphic for the Joker that works in the board update tool. As soon as a nice graphic for the Joker becomes available, I'll use it in my diagrams. Till then, I am using the graphic of the Guard with the letter "J", to indicate that this is a Joker.
And why doesn't the Joker inherit special abilities, but lose them if the opponent moves a different piece on his next move?
The Joker only inherits movement and capture. That's is a choice made by past inventors. I did not invent the Joker but I am clarifying it here.

If the opponent has promoted a pawn, the Joker inherits movement and normal capture abilities from the piece it promoted to.
I found another inventor of the Joker who implemented this point in the same way here:
http://www.talkchess.com/forum3/viewtopic.php?f=7&t=51616
I came up with a new scenario that still must be clarified:
White moves a pawn to the last rank and promotes the pawn to a Joker. Now it's Black's move and Black has a Joker. What is Black's Joker now imitating?
I'll give this a thought.

Just wanna know, in the position on the FP, if it is White to move, is Jb6+ legal?
Since in that case, the Black Joker will imitate the White Joker which is imitating the Black Rook, gaining its ability, attacking White's King.

Just wanna know, in the position on the FP, if it is White to move, is Jb6+ legal?
Since in that case, the Black Joker will imitate the White Joker which is imitating the Black Rook, gaining its ability, attacking White's King.
Jb6+ is illegal for the reason you mentioned here.

If the opponent has promoted a pawn, the Joker inherits movement and normal capture abilities from the piece it promoted to.
I found another inventor of the Joker who implemented this point in the same way here:
http://www.talkchess.com/forum3/viewtopic.php?f=7&t=51616
I came up with a new scenario that still must be clarified:
White moves a pawn to the last rank and promotes the pawn to a Joker. Now it's Black's move and Black has a Joker. What is Black's Joker now imitating?
I'll give this a thought.
Maybe it could pretend to be a pawn...

What if a Joker is on the 1st rank and someone moved a pawn?
It can move one square to the 2nd rank.
Why don't you just join the tournament?
He means if the white joker is on a8 and if black plays a6

What if a Joker is on the 1st rank and someone moved a pawn?
It can move one square to the 2nd rank.
Why don't you just join the tournament?
He means if the white joker is on a8 and if black plays a6
Then I call it the 8th rank. A Joker on the 8th rank imitationg a pawn has no legal move. Play with another piece when that is the situation.

I am going to add the next line to the rules:
The Joker must in that case imitate that type of piece of its own colour. This addition is important when, e.g., a Joker is imitating a pawn: when black plays a pawn, the white Joker must imitate a white pawn and not a black pawn.
The Joker in very short: Moves like last piece moved by the opponent.
Inventor:
The chess variant pages (Piececlopedia) mention: "invented by the famous chess problem composer T. R. Dawson".
It was also part of Superchess from Henk van Haeringen.
The Joker inherits the normal movement and normal capture abilities of the last piece that has been played by the opponent.
White to move, Black's last move was Rh8-g8. White's Joker moves and captures like a Rook.
Black to move, White's last move was Ne4-d2. Black's Joker moves and captures like a Knight.
In the same position, White to move: a move with a Rook is illegal, because that would make the Black Joker move like a Rook and give check.
In the same position, Black to move: a move with a Knight is illegal, because that would make the White Joker move like a Knight and give check.
The Joker must imitate a piece like it is of its own colour. This addition is important when, e.g., a Joker is imitating a pawn: when black plays a pawn, the white Joker must imitate a white pawn and not a black pawn.
- The Joker can deliver checks/checkmate as the piece it imitates, but it is possible to get out of check by moving a piece that causes the joker to not be threatening the king.
- The Joker imitating a pawn, can capture or be captured en passant
- The Joker imitating a pawn, can move two squares when the Joker is located on the second rank
- If the opponent has promoted a pawn, the Joker inherits movement and normal capture abilities from the piece it promoted to
- A Joker that imitates a Joker, inherits the movement and capture abilities that the opponent's Joker was also inheriting
The Joker does not inherit special abilities (but like any other normal piece it is subject to these special abilities)
- The Joker does not inhirit abilities to have side-effects on other pieces, like making transparent or freezing
- The Joker does not inherit the special way of capturing by the Time Thief
- The Joker does not inherit the royal function of the King and cannot castle
- The Joker imitating a pawn cannot promote
Whenever new pieces with new strange abilities are invented, it can be required to clarify how that piece works together with the Joker. The general rule is still: only normal movement and normal capture abilities are inherited by the Joker.