You're thinking mac-address-book type fields, right? Like where you can select the label for what phone number you're about to enter into a phone number field in an address book? Same/Simliar on iOS I believe.
It's a very cool idea in concept but I think you should reconsider how annoying it would be really to have to create my own form that I then fill in.
For various reasons i don't begin to understand, I routinely have trouble with the more complex interface things on my browser - drag and drop and fancy buttons can be a problem for me, I guess it's java cache related, I'm not sure. I can only imagine what people 2 versions back could have trouble with and then you'd have to make sure it worked on mobile devices.
Wufoo has a nice form and I think it's HTML5 - maybe HTML5 would help alleviate potential issues I'm alluding to in the previous paragraph. I think there are a few companies out there doing contact forms and you'll probably do better to find one that is not WordPress specific - they'll give you embed code and you can go from there. I've even seen a bunch of them in the Chrome web store, as well. (As a last resort, google for alternatives to gravity forms or wufoo)