Will a step nose expand with humidity, or is it considered fixed?

Good morning everyone!

I’ll soon be installing a T&G red pine wood floor and have planned for an expansion gap around the perimeter of the room. On one side, there’s a step with a wood stair nose.

I have two questions for you:

1. What’s the best way to install the stair nose? Should I glue it down and use finishing nails, or screw it from underneath? (I do have access to the subfloor from below.)

2. I was planning to install the floor planks tight against the stair nose, without an expansion gap, assuming the nose itself can expand. But depending on the answer to question #1, will the stair nose act as a fixed element (like a wall), and therefore require a gap as well, or not?

Thanks in advance — I’m looking forward to your advice!