Re: engagement ring
I wear both my wedding and engagement rings every day. I had them fitted to my finger and they are the 'euro' shape and they are super comfortable. Even when working in the kitchen, I only take it off if I have to get my hands into food for mixing with my hands etc.
Peppers - you should take your ring in to a good jeweler. The diamond should never come off like that. My center diamond is set in a tension setting (not embedded, but held by prongs only), and I have worn it swimming, rock climbing, paintballing and numerous other things and have never had to worry. You should get your diamonds reset.