Hi. If I were to purchase the DLC here, would I get a steam key and be able to play it there as well?

Of, course. Happy to help.

I don't know if this is too late or anything, but the same thing happened to me. The Download Now button at the top and the Download or Claim button under Download didn't work. However, I use the app on my laptop, and clicking the Buy Now button at the bottom right corner of the screen worked for me.