I thought that scene worked great. It showed that:
Given how lonely Clint felt at the time (him missing out on his family because of what he did as Ronin), I don't see it odd at all for Clint to open up to Kate. Kate isn't really that unknown at this point:
1. She and Clint have gone through a few scrapes together at this point.
2. Clint has begun to know a fair amount about Kate and her family.
3. Clint understands the trouble Kate is in.
4. Kate has shown herself to be a person who could sympathize with Clint and has made a pretty good effort to cheer him up.
With regards to Kate feeling for Clint and Natasha, I think that is only basic human empathy/emotion? I mean, how many times have you felt for any movie character you've just seen for the past hour? You barely know that character (you only started watching the movie an hour ago) and yet that fictional movie character can evoke an emotion.
In this case, Natasha and Clint are very real people to Kate (she isn't just watching a movie), she's been with him the past several days (not just 1 hour like characters in a movie) and she (and the world) has probably known the rough history between them by now. There's an Avengers musical to show this, and every one knows the Black Widow was dead.
I do agree the action choreography could have been better in this one. I guess you can't make them all good (like last week) on a TV budget and schedule.