Is this a checkmate?

no

Your question belongs in the Forum for Beginners, not in the Endgame Study Forum. If you or your sister can't see that this is not checkmate you will need to revisit the basic rules of play. Exercise the walks of the different pieces in a chess set and you will see why. Hint: there are 2 reasons why this ain't checkmate.

No, but checkmate will come soon.

All right thanks guys!

Also, there is an illegal white pawn on the black side of the board.

Maybe this is the position he talks about 

This position is not mate yet because of Rxf4!, Taking a rook is a great move because it's the only good move that prevents future checkmate and at least winning some material (a rook for queen and a pawn)

Kxh3?? Is a blunder because of Rxf2, a3, bxa3, and then black is in a situation called zugzwang (it means a side must make a losing move) and forced to move to h4 when the rook is ready to deliver checkmate with Rh2# checkmate

@Raphael

Does "zugzwang" really mean the side to move must make a losing move?

In any losing position that is not already checkmate the side to move can make only losing moves and in any position that is not already checkmate where all moves are losing the position is losing. (In fact, it's also vacuously true that all moves are losing if a player is in checkmate, but then the player would be lost rather than losing; also presumably not in zugzwang.)

In that case, wouldn't it be simpler to just say "losing"?

(Also, I'd call 1...Kxh3 a bad move rather than a blunder. It shouldn't affect the game result, it's just a less robust defence.) 

It's obviously a White win whatever Black plays. How can you say it should be a victory for Black and then say you bet on a White victory?

2.Qe5 is only one of several - here's 2.Qc3 against SF, for example.

to avoid black getting checkmated, take the white queen and rook off the board and check white with blacks rook

?

Easier to take the black king off the board - that should do it.

Almost ... it means the side to move is put into a worse position because they have to move, a worse position than if chess had the option to sit still with no moves and have the opponent make the next move. It's a common thing in endgames, where sometimes it would be impossible to force a checkmate if the lone king could sit still without making a move. 

That's what I thought it meant.

Black is not in zugzwang after 1...Kxh3 2.Rxf2 a3 3.bxa3 - he's just losing (with or without the move).

No

In game situations that is exactly right. In compositions the definition is more generalized. Like, when the duty to move changes success to failure you are considered in zz.

But don't ask me why SF played 2...a3. I kibbitzed it after (with the same SF) and it never made it higher than third option.

I found 2.Qg2 quicker (which doesn't necessarily mean it is).

Edit. The difference is more down to me wasting time pushing the h pawn in the first example. I was worried about finishing up with a drawn queen v rook's pawn, but it's not really an issue. Just pushing the b2 pawn to promotion finished up 10 moves quicker (I've added a variation to the first example). That is to say, not much fantastic at all, really.

@BlackaKahn

Actually, it's not exactly what I thought it meant.

You say 

it means the side to move is put into a worse position because they have to move, a worse position than if chess had the option to sit still with no moves and have the opponent make the next move

My understanding is that a position is a zz not if chess had the option to pass, rather the player on the move had that option in the position under consideration only. 

The idea of zz is useful in chess in contriving to arrive in such positions without having the move,  but chess with a pass option is a completely different game. This would simply be a draw for example. Would you count it as a half point zz?