This means the story part is finished, right?

According to the release post, yes

The Caves of Qud 1.0 update includes:

• The conclusion of the main quest, with multiple endings

Old save won’t work, gotta start fresh

And it’ll be worth it, depending on how long it’s been there’ve been a lot of changes and improvements

I know it’s on the “other” site, but this post from a few months ago has several people saying they got it working quite nicely: https://www.reddit.com/r/cavesofqud/comments/1dcqyym/regarding_qud_on_the_steam_deck/

Works great on Deck!

It’s truly worth the price. DM me your steam handle if you feel comfortable and I’ll gift it to you.

Yes, it works fine on the Deck! I find the controls less convenient than kbm, but that might just be me; everything you need is mapped and it feels quite intuitive after a bit.

I was reading that the devs specifically did a revamp to map the deep control scheme for the Deck. So the answer to your questions is most likely yes.

I’m playing it regularly on my Steam Deck with no issues. Just took a quick read through of the control mapping.