Yomping!! First of all, big hugs. Your little one definitely is quite challenging, please don't think you're abnormally unable to cope or anything like that! We all would be at the end of our tether.
I reckon what you need is A PLAN. A plan for what to do and how you and DH divvy up your roles in relation to DD. Be clear, plain and honest with each other about how her behaviour/challenges affect you and talk out what would be the best way to deal with them.
Maybe I'm just a super-weirdo but I always find that things get so much better when you have some kind of a plan in place, even a very basic one, even one that just address how YOU will react to all the craziness that can get thrown at you. Ie, when crazy shit happens that I don't know what to do to fix, I will do X and that will make me feel more in control.
Just throwing ideas out here, but maybe - for example, if DD cries randomly (ie feeding, cuddle, change, attempt to put to sleep doesn't work) for more than 15 minutes, you put her down, go and take a breather for 5 mins (time on your phone). And make it a policy to strictly enforce it. You know what I mean? I have so many of these crazy little schemes I have in place (about everything) to cope. And I make lists. Lots and lots of lists.
I do this with DS. When he really loses it and has been crying for 20 minutes + and nothing I do can fix it - I swaddle him in a big cotton muslin, I go into the other room and check my emails/have a breather. Half the time when I go back he's calmed down or is asleep. The other half he's still crying but I feel better about picking him up.
Also I don't think there would be anything wrong with trying out Ferber method/controlled crying for a week or so. Or whatever variation on that you think you'd be comfortable. DD won't remember it, and you can find out whether it will improve things. If it doesn't - hey who cares, it's only a week.