Some of these ideas are pretty good. And then often also very old. (Which probably should not be a surprise.) Promotion by capture was used in the large Shogi variants (on 17x17, 19x19 and 25x25 boards) from the Edo era. I designed several chess variants based on this; the most Chess-like (which can be played with an ordinary Chess set on 8x8) is Veteran Chess. What you called 'Parasitized piece' I usually call 'contageous', and in the 19x19 Shogi variant the two strongest pieces were contageous. This is a good mechanism for preventing these pieces are traded out of the game. I used this principle in my own chess variant Werewolf Chess. A Swapper / Kicker occurs in Odin's Rune Chess (the Valkyrie). It can swap with friendly pieces, or kick those back to any square of the path it followed to reach those. Mats Winter invented 'Catapult pieces', which can sling other pieces over the board. The King in orthodox Chess actually can be considered long-tailed piece: the only move where it has the opportunity to leave a tail is castling, and castling through check is then forbidden by the rule that you should not expose your King to (e.p.) capture. In the variant 'Caïssa Brittania' the royal piece is a long-tailed Queen that cannot pass through check. Something similar to abdication occurs in the variant Tamerlane II, where it is called 'Succession'. It takes a turn to swap the royal King with a non-royal one. The difference is that you are only allowed to do this while in checkmate.
Unorthodox piece ideas?

Flip a coin to see if it's black or white - one horse is a cow instead, and instead of moving in an L shape it moves one square diagonally once and then just sits there. It doesn't move and can't be captured. It's like there's a dead square on the board. 🐄
Just off the top of my head as you may be able to tell.

I want a simple Queen Fairy (Queen Ferz). Just a Ferz that can also move two squares orthogonal if the square in between is empty.

I want a simple Queen Fairy (Queen Ferz). Just a Ferz that can also move two squares orthogonal if the square in between is empty.
You might like Fairy-Stockfish, which lets you create custom pieces.
Fairy-Stockfish Discord Server: https://discord.gg/3aSyWkxn

Have you heard of Enigma Chess? Enigma chess is a chess variation that incorporates a new piece, the enigma piece. Each player assigns his own chosen power to one enigma piece and both players then have corresponding pieces. More details on this in the thread . . . But here is a big question: Is the classical chess board setup flawless? Here's why I ask, and the answer may not be black or white, but both black and white: Why should white's kingside castling be to the right, but black's to the left? I see an unevenness in the game here. But if kings faced queens, both sides would castle perfectly even. It's not black or white . . . If two right-handed, or two left-handed players play, why should kings not face queens, and kingside castling be perfectly even (to the same side from each player's perspective?) Here are my thoughts on it: Let's say two armed men, both right-handed, or both left-handed, face each other, and both are carrying both a shield and a sword. When they face each other, will swords face swords, or will swords face shields? But let's say a left-handed man faces a right-handed man, will swords then face shields, or will swords face swords and shields face shields? Or if two couples, two kings and a queens, walk up to one another, both having their partner on the same side, when they face each other, will kings face kings, or will they face the queens directly? Now why one cannot just shoot down what I'm asking about is because a game like chess involves two people. A car resembles a human in many ways because it is designed to be used by a human and thus needs to mirror many things the human driver's body performs. So a car has 'eyes' (lights), a 'mouth' (the horn), a 'heart' (the engine), 'blood' (the fuel), 'feet' (the wheels), 'eyelids' (windscreen wipers) etc... In the same way, chess is a reflection of the players. So if two right-handed, or two left-handed players play, why should the classical board setup not be as mentioned earlier, in view of these arguments? But if a left-handed player plays a right-handed player, I'd leave it just as it is. Yet even so, black should still have the chance to move first. Hmmm.... Your thoughts?

I want a simple Queen Fairy (Queen Ferz). Just a Ferz that can also move two squares orthogonal if the square in between is empty.
This seems like a pretty conventional piece. A lame Dabbaba combined with a Ferz.

Have you heard of Enigma Chess? Enigma chess is a chess variation that incorporates a new piece, the enigma piece. Each player assigns his own chosen power to one enigma piece and both players then have corresponding pieces. More details on this in the thread . . . But here is a big question: Is the classical chess board setup flawless? Here's why I ask, and the answer may not be black or white, but both black and white: Why should white's kingside castling be to the right, but black's to the left? I see an unevenness in the game here. But if kings faced queens, both sides would castle perfectly even. It's not black or white . . . If two right-handed, or two left-handed players play, why should kings not face queens, and kingside castling be perfectly even (to the same side from each player's perspective?) Here are my thoughts on it: Let's say two armed men, both right-handed, or both left-handed, face each other, and both are carrying both a shield and a sword. When they face each other, will swords face swords, or will swords face shields? But let's say a left-handed man faces a right-handed man, will swords then face shields, or will swords face swords and shields face shields? Or if two couples, two kings and a queens, walk up to one another, both having their partner on the same side, when they face each other, will kings face kings, or will they face the queens directly? Now why one cannot just shoot down what I'm asking about is because a game like chess involves two people. A car resembles a human in many ways because it is designed to be used by a human and thus needs to mirror many things the human driver's body performs. So a car has 'eyes' (lights), a 'mouth' (the horn), a 'heart' (the engine), 'blood' (the fuel), 'feet' (the wheels), 'eyelids' (windscreen wipers) etc... In the same way, chess is a reflection of the players. So if two right-handed, or two left-handed players play, why should the classical board setup not be as mentioned earlier, in view of these arguments? But if a left-handed player plays a right-handed player, I'd leave it just as it is. Yet even so, black should still have the chance to move first. Hmmm.... Your thoughts?
I really don't know if radial rather than mirror symmetry chess has been studied. I can't think of anything too egregiously wrong with it.
Stockfish seems to not understand how castling should work in that position. Which leads to some confusing behavior. If it's on the side which should castle normally, it really likes queenside openings. If it's on the side which it misinterprets as being able to "long castle" one square or "short castle" three squares, kingside moves are preferred on the first move.
If I GET RID OF castling entirely in order to solve this, Stockfish has very different opinions about the position. It refuses to even touch the Kingside pawn and has ideas about moves that literally never see play in normal chess with castling. So I think sadly stockfish will not tell me much about this. I will say the "no castling" variant is remarkably even. Although the openings appear written by an alien being.

@pds314 Hey! Thanks a lot for your reply. I appreciate your insights on this. I wrote some further details about enigma chess on the forum article I created under Chess News, "Have You Heard About Enigma Chess?" If you'd like, you can check it out. And once again, all insights on it are appreciated.
These are some ideas for pieces that would do something beyond just move or capture in a different pattern.
1. Capture promotor: promotes to something else when it captures. Could be any capture or only certain ones.
2. Piece spawner: summons a certain piece at its location and immediately moves it.
3. Swapper. Swaps positions with a piece instead of capturing.
4. Kicker: moves a piece in some way instead of capturing. Could result in that piece capturing or making moves it otherwise couldn't.
5. Wololo: instead of capturing pieces, it can color swap them at a distance. Very OP ability.
6. Parasitized piece: Any piece which captures it promotes to it.
7. Infectious piece: Any piece which it attacks after its move, promotes to one. For extra chaos, make it any piece friend or foe.
8. Heir: A piece which is not royal, but will become royal should the king abdicate. The king cannot abdicate in check. An abdicated king loses royal powers but remains on the board as a Mann.
9. Regent and boy king: Both the regent and the boy king can move on the same turn, but both must move for either to move. The boy king is royal and the regent is not.
10. Demon. Pieces captured by the demon are banished to the underworld. A separate board. If the demon is captured, it is banished to the underworld. If a piece would capture another piece in the underworld, the capturer escapes the underworld and is summoned to the board in the corresponding location, capturing the piece that was there. If this creates an infinite loop by capturing a demon on a square that's occupied in the underworld or something, the demon survives and the other piece does not. Royalty sent to the underworld are considered checkmated if they remain there after the turn resolves.
11. Vehicle A piece which can be mounted by another piece and move with that piece attached. That piece can also dismount and move normally. Could be immobile if not mounted.
12. Long-tailed piece. Whenever it passes over a square it could land on, it risks en passant on that square the next move. Not just by pawns but by any piece.
I think there's definitely some interesting variants that could be made using these concepts.