You will be fine without it if it’s indoors.

If you wanted to strengthen the wood and protect it from getting dinged as easily a couple coats of boiled linseed oil will help. Me personally though I’d just leave it alone, I doubt it’ll get enough abuse to warrant the extra time.

I'm not even a woodworker really, but ... No. It's indoors, it's not even in the bathroom which is basically the only place you have to worry about humidity.

If you really want a cheap preservative, you can use mineral oil. I've restored the handles of most of my garden tools with just sanding and mineral oil, and they've stayed quite sound even being outside year round. I even did a raised garden bed like that three years ago and it doesn't show a spot of decay, even the parts covered by soil. And it gets watered three times a day so it's very wet.