Texture and cutting of avocado and Mango used in sushi
Avocado: Soft, but not too soft.
Mango, a little bit under ripe.

So, it is always to your own taste.

As to how to cut the mango, I would cut in half and make thin sliced (a little bit like cutting fish for sushi).

Video showing how to cut a mango (like a pro) for sushi rolls.
https://www.youtube.com/watch?v=NL6J_hy3sHM