How should I land summer internship?

Hi fellow monkeys,

1. In short we won a competition (team of 3) like a week ago and we have an upcoming insight day at a midsized firm, which also includes a lunch at a restaurant with an MD (also he is the one who will be with us all day). My two teammates don't really want to land an internship, but I do. Could you guys give some advice on how to ask the question 'I would love to spend the summer working for you', without being pushy or annoying? Should I ask this at the end of the insight day, or during lunch? Also, In our country interns usually don't get paid, so I would also love to include that I am willing to work for free, but again, without being pushy.

2. Also the three of us will attend a week long insight week at a midsized firm (it's an investment firm with a retail divison too), in which each day we will be introduced (shadowing, talking to brokers, traders and stuff) to each desk of the S&T division. We got a detailed program for each day which ends at 3PM. So my question is: how can I make sure I will be able to stay longer, show my enthusiasm, and secure an internship for the whole summer?

Thanks big time!