A pair of interesting puzzles

Actually, Toadofsky is right after all. In mrwatson1's position, it's still checkmate even if both of the knights are taken out. In my position, the rook is essential to block the flight square of the king. So, so far my solution still holds the record.

@Patralgan,
One of the pawns that are giving check can be removed and it'd still be a checkmate. (d4 or f4)

I've considered this. A checkmate which contains a double check cannot be a solution because one of the checking pieces can be removed and it'd still be a checkmate.

Well, I was fearing this...

I've spent some time trying and gave up. I tried to use a pin to increase the number of pieces, but it was not working...

But I decided to give a last shot:
lichess.org/editor/8%2F8%2F2ppp3%2F2pkp3%2F2Ppp3%2F1P6%2F8%2F8%20b%20KQkq

What do you think?

Lichess editor doesn't update the position in the textfield. The position I wanted to show is:
http://pt.lichess.org/editor/8/4K3/2ppP3/2qkq3/2Pp4/1P3P2/8/8%20w%20-%20-%200%201

This is about the best I can do:

8/8/8/1PPk1P2/1PP2P2/1PP2P2/8/8 b KQkq

9 pieces, or 10 if you include the White king to make it legal and 11 including the dead meat in the middle. ;)

@okgo
The pawn on e6 can be removed and the position is still a checkmate

Interesting ... If we remove e6 both kings will be in check. but the black king will still be in checkmate.

@Ashura That's my point. If you agree that one cannot be in check in order to give checkmate, the e6 pawn is needed to/is contributing to the ckeckmate, and counts. But it seems the rules don't care about this and allow a checkmate while in check. It's strange because since it's black to move it could just capture the white king and win. Perhaps we could rewrite the rules so that the solution fits and makes this a good exercise.

@okgo
The position you showed is not checkmate as black can play Qxc4.
Here's a slight modification to your answer to get a checkmate with 12 pieces.
http://en.lichess.org/editor/8/4K3/2ppP3/R1qkq3/2Bp4/1P3P2/8/8%20w%20KQkq

@okgo
Yes I think maybe we should add a rule that removing a piece shouldn't result in both kings being in check.

@TheCombinator
If you remove the queen on c5, it's still a checkmate.