K-G vs K

This is not primarily a 50-move issue; even in complete absence of such a rule short-range pieces tend to lose their mating potential on large boards. On large board it doesn't make much sense to keep the allowed number of moves as low as 50 anyway; this would spoil many forced wins. In orthodox Chess this number was picked because if the win can be forced, it can almost always be forced within 35 moves, so that you did not make progress for 50 moves is a very strong indication the game is indeed a dead draw.

The point is that neither a King or a Guard alone can confine a King. So to prevent it running to safety you always needs the pair. But the pair moves twice as slow as the bare King, because you are not allowed to move both in the same turn. So the bare King continuously outruns the pair. That doesn't mean it is always hopeless: the pair covers an area of a certain size, and the bare King would have to encircle that to get to safety. The total width of the covered area can be 6 squares. So the bare King has to gain 6 steps on the pair, and each 2 turns it only gains 1 step. So it takes ~12 turns for the bare King to overtake its persuitors, in which he can cover a distance of 12. If he runs into an edge before that, he is confined (and then gets trapped against that edge, so that trying to outrun the K+G is not the best defensive strategy). But if the edge is further away, he then moves towards freedom once he has left the pair behind.

So the defensive strategy against a set of short-range pieces is to use the board as a race track, running circles along the edge to get clear of yoiur pursuitors, and move towards the center when you get ahead of them (or have dispersed them, so those near you do no longer form an effective unit).

A Rook can confine a King virtually without moving (you have to switch now end then to the opposite edge when you get attacked), so there is no way to outrun it. The strong side can use nearly all his moves on his King, to approach it and confine the bare King in the other dimension.

A similar situation occurs in the 4-men ending King + Bishop + weak piece vs. King. Unlike the Rook, the Bishop cannot confine a King. But King + Bishop together can. The Bishop hardly needs any moves for this confinement; it covers a diagonal, and the attacking King has to cover the square where the bare King could sneak through. So you basically have a stationary Bishop and King vs King, the latter of course having equal speed. So the attacking King can just keep up with the bare King. The point is that the bare King cannot reverse its direction without losing a tempo; to maintain the threat of crossing the diagonal, it has to keep moving in the same direction. Where it sooner or later will encounter an edge, so that it has to reverse. At that point the strong player has a 'free move' to approach his weak piece. So sooner or later that weak piece gets to the action, and with the aid of it wou will be able to drive the King away from the diagonal, and then tighten the Bishop noose. So this end-game is a forced win on any size board, but it can take excessively long.

As to board size: that I mention square boards does not mean it cannot be done on rectangular boards. Just that I never tried it. Obviously when you can do it on an NxN board, you can also do it on NxM or MxN with M < N. But if you cannot force mate on NxN, you often can force mate on MxL, with M<N and L>N. E.g. on 15x15 KGK is almost always draw, but on 8x100 it is a very easy (but very lengthy) win. The point is that you can cut confine the bare King along the narrow dimension, and then slowly but surely push him to the edge along the length of the board.

How does guard move