How should I begin my meeting with my former teacher whom I haven't been in contact with for three years?
If she has already agreed to write you a recommendation letter, I would think the most nerve-wracking part is over. At least for me, meeting with professors and not knowing with 100% certainty that they would write a letter for me was the source of most of my anxiety over the meetings.

I would definitely not ask if she brought you a souvenir from Japan (or anywhere else) unless this is a culturally acceptable thing (at least in the US, such a question would likely be considered strange at best and rude at worst). I would also avoid questions/topics that can be considered personal, like her living situation, love life, et cetera.

I think it would be fine to ask her about her travels (e.g., "tell me about Japan"), or ask her about her current research or experience at the school. At least to me, those are innocent questions that also show that you have an interest in what she has been doing. From there, you can lead the conversation to what you have been doing and what you plan on doing in graduate school, and this naturally leads to a discussion of the recommendation letter.