An easy lower bound would be to set a White queen in a corner and set up an easy mate in 2.
[FEN "Q6r/8/8/8/8/8/4k1p1/r6N b - - 0 1"]
This sets Black down to a minimum of a king, two rooks, and a pawn in count and value.
There are three chess variant GUIs supporting both atomic and king of the hill that I am aware of:
The latter is well suited for automated engine matches and human vs. engine games, but less for game analysis. All three should work fine with the multi-variant Stockfish you downloaded.
I haven't heard of that definition being used for a major piece, but it's a neat observation. Such a piece (call it X) should be able to force a checkmate in a K+X vs K situation.
Say White's pieces are on d3 and e3, while Black's is on d5:
1) White plays Xd4, and Black's king must retreat. Say ...Kd6 is played.
2) White plays Ke4. Now Xd5 is coming, ...
This piece is widely used in different fairy chess variants. It appears under several names, including Commoner, Guard and Man / Mann.
While the table in the Wikipedia article also linked to in @Allure's answer states it's worth about four points, the article itself limits that value to the endgame:
In the endgame, where there is usually little danger of ...
Yes, that's a "square". The word "cell" is never used. E.g. from the Wikipedia article:
Chess is a two-player strategy board game played on a checkered board with 64 squares arranged in an 8×8 grid.
The 64 squares alternate in color and are referred to as light and dark squares.
Or, if you prefer a dictionary:
3 : any of the quadrilateral spaces ...
That is an interesting distinction theory. Although this king-like piece could deliver checkmate with the help of the king, I've always considered the difference to be based on the value of the piece. A king-like piece would only be worth the same as the other minor pieces.
The king is worth about 4 pawns if its loss didn't lose the game, which means it's neither a minor or a major piece, but something in between.
(Major pieces like the rook is worth 5+ pawns; minor pieces are worth 3.)
This seems to just be a version of Chess960 restricted to certain openings to make castling more like "normal". Thus the positional considerations should be identical to regular chess, and regular chess principles (e.g. control the centre, develop your pieces early, gain space, etc.) should apply with equal force here.
What does change is the pattern ...
Apart from the main chess sites (lichess.org, chess.com), there are a number of other sites where variants can be played:
Based on lichess's source code and powered by Fairy-Stockfish. The list of supported variants is huge, including popular ones like S-Chess, Capablanca, and crazyhouse and 960 variants of these, as well as xiangqi, ...
As SmallChess notes, the rook pawns would be able to capture on the first move. That in itself wouldn't be a problem, since the player can just capture back, but then we have a situation with two half-open lines where the pawns are pinned against the rooks behind them. That's a lot of tension already after the first move, and on the flanks, which is quite ...
It's not a sacrifice, it's a trade after 1. Gxc2 G*b3 and taking the elephant on a3 next.
On the other hand, since doubutsu shogi is strongly solved, you can use an engine/tablebase (e.g. https://github.com/fuzxxl/dobutsu) to set up and see the solution anyway. (I haven't done this, though.)