Now that I've (hopefully) figured out the route for the trips into town, let's go back to where I was in my speedrun planning. I just finished figuring out the fastest way through the Emily/Jane conversations at the start of the game.
After talking to the women, you can go into town or go to Topham's. There is a piece of paper on the way to Topham's, about the jeweler. Also, there is a book that mentions the trivet, at Topham's. Since Topham's area has two triggers for things in town, and town has no triggers for things in Topham's, it makes sense to visit Topham's first.
The first thing you do in Topham's is find the toy mouse. It is either by the carousel, under the table or under the chair. For me, it's always under the table.
The first conversation with Topham is a little long to get through. As in, there can be five places, where you pick different options. There are some consistencies, like "if you ask about making objects move, he starts a new conversation about his students". The fastest way is to select the second option each time.
The first conversation with Topham always ends the same way. He offers to give Nancy a lesson, she turns him down, and he gives her a logic puzzle. There are three places in the conversation where you can make choices, and his response is the same, no matter which option you pick. With all three, the first option is shorter, so it's faster to pick.
So, the Topham conversation is second option three times, then first option three times.