The first time you go to Topham's house, you have to read Josiah's notebook all the way. This triggers the trivet puzzle, and it lets you ask Emily about her mother's middle name. Getting the middle name is simple. Go to Emily and ask her, then finish the conversation immediately afterwards. It kind of stinks that you have to make a separate trip to Lilac Inn just for that, but there's no way around it.
To get the mirror from Topham's house, you have to do a simple matching puzzle. I don't think there's a way to speed it up, besides "have the answer memorized / written out ahead of time". Same thing with the slider puzzle, guarding the mirror piece in the den at Lilac Inn. Both of those puzzles are different on Junior and Senior Detective modes, although it's not like one mode is significantly longer than the other for this puzzle.
Is the clock gear puzzle at Jim Archer's the same on both modes? I think it is. The pieces always start in the same spots, so that makes the puzzle a little easier.
The first Jim Archer conversation is easy to navigate. On the first choice, ask if he's Jim Archer, which is about .4 seconds faster than the other option. On the second choice, avoid talking about Emily, as that leads to an aside about Emily before Jim turns the conversation to the economy. On the third choice, don't mention the depression, or Jim will give an aside about how that talk is pointless, before he gets a phone call, and you can end the conversation.