Sorry, but a design on every block and wall that is clearly Majora's Mask in Termina means a lot more than a pic on the side of the wall of the Spirit Temple really looks almost nothing like Majora's Mask at all. It's an interesting design, but its not Majora's Mask. If the Gerudo used the mask or it was a part of their history, I'm sure we would have heard about it in OoT, or at least in MM itself. But we didn't, so there's no relevance here.
And as for the Happy Mask Salesman and the mask being from Hyrule, there's also no in game evidence that points to this either. When your talking about something like this, something that the designers of the game make no mention of at all, but there are hints towards it, you have two overall options.
First, you have the option that the plot isn't important at all, so they make no mention of it. If, and thats a really big IF, Majora's Mask was from Hyrule, they would have mentioned it. There would be no reason for them not to. Unless....
Second, its an obvious thing, so they don't bother going into detail about it. It is quite obvious that the Mask is from Termina, as that is where the game is set, there are various designs throughout Termina, and many of its townsfolk have details relating to Majora's Mask in their history. Nothing at all points to it being from Hyrule, except you saying that the HMS is from there and he had the mask.
I think its a combination of the two reasons above. The designers put the game in Termina for a reason. They wouldn't have the main boss/antagonist of the game be from Hyrule and not mention it. That would be, just stupid really. The Happy Mask Salesman said he searched far and wide for the mask, and in the cutscene where he gets knocked out by Skull Kid and has the mask stolen, he is in the section of the Lost Woods at the beginning of the game; The section that is obviously very close to the boarder of Termina. So it makes sense that the HMS went to Termina, found the mask, and was on his way back, when he got robbed.