How to make this stair rail less slippery

Painting with a roller leaves an "orange peel" type texture that you can't match exactly with a brush. When you brushed your new paint on, the orange peel texture in the unsanded areas just telegraphed through your new brushed paint, but the smoothly sanded parts had no such base texture so they look visibly smoother.

To fix this I'd go over your patches again with a small roller and you should get the same sheen and texture as in the original paint.

Make sure everyone using the bathroom uses the fan.

The exhaust fan has some how moved. Inspect it to see why and check to see if there are any issues with the duct. Also make sure it is sized appropriately for your space. See this guide on bathroom exhaust fan sizing from the Home Ventilating Institute. You may want to consider a larger fan (or better fan) if it's not working.

They have a number of different options for fan switches that have timers that keep the fan running even after the lights have been turned off. I've even seen one with a humidity sensor that automatically turns the fan on and off. Consider getting one of those.

You're going to want to repaint. In a bathroom that means you have to thoroughly clean the walls whether you have mold or not. Too much stuff accumulates on walls in bathrooms to get good adhesion without cleaning. That might be why you're seeing peeling more so than from the steam. Scrape off any peeling paint and sand the edges to feather them and repair any large dips or gouges.

Paint with a high quality paint meant for bathrooms. Most good paint these days seem to have mold inhibitors in them. You don't need to go with a higher sheen paint. In the past I've used Zinsser Perma-White, which I believe was one of the first paints with mold inhibitors. It is available in eggshell. I recently used Benjamin Moore's Aura Bath & Spa paint which is in their Matte finish. It's somewhere between flat and eggshell, maybe leaning more towards eggshell. Happy with both but really thrilled with how well the Aura went on and has been holding up.