It says "they held finger-tutting contests all night long," I'm not exactly sure where you're getting "they did it" from. So if it's ambiguous somehow, let me know!
No worries, I was thinking "Oh, maybe this is against site rules or something." I really like the artist's comics/illustrations so I wanted to do these as well for the reason you listed.
Yes, most Japanese people wear their wedding ring on their left finger. Kazuma noticed the ring he gave her from day 50 is on her left finger here. He was asking if there was any special meaning "here" because he's in a different world. It was entirely possible for her wearing it on her left...
When I was going through this page, I thought to myself "this really seems like a proposal, not him asking her to just be his girlfriend." The translation is pretty direct here as he is asking her out, but Japanese language is based very much on context as well. I'll be upset if it gets to day...