This feels like a thread for me.
When having to pick a small natural number that don't need to be precise, I have a preference to pick primes. Example: volume control, number of nuts to eat, seat number.
When having a pick an arbitrary large number, I tend to go for one of the known constants. There are a lot of constants, so it's easy, no matter what magnitude. Orders of finite sporadic simple groups, digits of Feigenbaum constant, coefficients of j-function, etc.
I like irregular plane tiling, like Penrose tiling and pentagonal tilings, I will pick such pattern if it's available.
2) Number theory and algebraic geometry. But I don't limit my interests to just those.
3) I used to make sure I step on each square tile on the ground once while I walk, and never on the boundary, but eventually stop because it's not practical.