The Carpenter is open, just like the rest of the shops, Monday through Saturday from 6am to 6pm. Ken is in charge of the shop. On Sundays he can be found wandering the village next to his shop.
At the beginning of the game Ken will supply you with an ax and a hammer. Later in the game he will upgrade your hammer to the Super Hammer. Once you have him built you your first house expansion, return to his shop once it's done to receive the Super Hammer.
Ken will also built you other things besides your home expansions. He can built a Bridge so you can reach the other part of your crop field, your Hot House, and your Sheep Barn. When you request something to be built, Ken will arrive at your house the next morning to tell you how much it will cost you and how much wood he needs. You can either tell him no, or let him know he can proceed. If Ken is building something for you, you can check how long it will take him by looking at the Schedule in your system notebook.
If you wished, you could even exploit Ken. The Q&A section of your system notebook hints at this a little bit. If you check the tv and notice you're expected to have bad weather the next day, head into town and ask Ken to built you something. The next morning instead of bad weather you'll have Ken waiting for you at the front door. Just tell him you changed your mind about building something and he'll return to the village.