There are lots of hints related to this. Play the piano 3 times before taking you-know-what and you'll see that there is a common theme: sheep, lamb, farm. Play the piano again after taking you-know-what and it gives you an important clue. Examine you-know-what and the mat and they have something in common. And, of course, you know what you need to get past the dog. Also, if you missed all these clues, you should know what you need to unlock a locked door - only that doesn't work. Try moving the mat in various ways.
Viewing post in Igor's Quest is ready for testing
Yes, you're quite right; the clue of the locked door, and then examining the mat, were what helped me connect the dots to solving the puzzle at the mayor's house. And once I had the meat, it was obvious where it needed to be used. The only thing that puzzled me (not important for solving the game, just my personal satisfaction) was WHY solving that puzzle resulted in meat appearing, although I did figure it out when I remembered that earlier comment. I even went back later and examined the piano again after you-know-what, to verify my idea, but I just didn't think to try playing it again.
Another hint was at the stable. This is the location adjacent to the dog. If you examined the stable, you got told it was the mare's house. 'Mare's house' sounds like 'mayor's house'. Get it? That's why I added the horse. The stable was originally empty. If you examine the horse, you'll see that she's a mare and the stable is her house.
The mat puzzle was inspired by one of my most memorable puzzles of all time - the tea remover in 'The Hitchhiker's Guide to the Galaxy'. I laughed out loud when I first used the tea remover and the rabbit turned into a rabbi. (Get it? Tea-remover = T-remover. It removed the T from rabbiT.) As my game is oriented around word play, I thought of having a similar puzzle, then came up with the idea of adding a letter, rather than subtracting a letter. But how do I add a letter? That's where the piano came into the picture. Pianos have keys and each key has a letter on a musical scale from A to G. I drew up a long list of words where I could add a letter from A to G to create a new word. One particular word jumped out at me. That lead to the dog.
The mat was originally going to be at the castle, but that didn't feel right, as there was no reason to have it there. When I was populating the village, I thought of adding the mayor's house. Now I had somewhere to put the mat. I have played a few adventures in the past where a piano key turned out to be a door key. I did a few things to suck the player into thinking that was the case here. But it was a bum steer. The key actually had a different purpose, as you now know.
Got all that? As you can see, my design process is sometimes quite convoluted. Lots of really good ideas got discarded along the way as I didn't want the player to be completely bamboozled. I originally had anagrams, palindromes and lots of other verbal trickery, but this would have been too hard to implement with the two-word restriction.