Mezuzah questions

Shavua tov, everyone. Hanging up mezuzot and I have some questions.

The stairs in my house have a proper doorway, doorposts and a lintel and everything. Not sure if a mezuzah is needed here since it leads to the stairs, not a room.
The door to my basement is located inside a bathroom. There is space enough where I can put a mezuzah inside the doorframe, meaning it’s only visible if you open the basement door, so if the door is closed it’s not exposed to the bathroom or anything. But that feels iffy to me since I know you don’t put them on a bathroom door.
I don’t know if my closets count as walk-in closets or not, and I’m a little confused with all the talk about amots when I look it up, plus there seem to be varying opinions anyway. One closet is 101” x 30” and the other is 75” x 44”.

Thanks for the help!