Oh hon (((hugs))) - it's hard isn't it? I know it doesn't help you now but it does get easier, it really does. At 5 weeks I don't believe you can spoil your dd, she is so little, and of course she was wants a lot of picking up and cuddling. Men, eh? Stick to your guns and do what feels right to you, there's time for leaving her to cry for a bit when she's older - she's only just out in the world!
There is no answer for the guilt you feel for your first - you can't give him the same attention you did before, hsi little world has turned upside down and it's no wonder he's more clingy and attention seeking. The way I reasoned it was that while I had taken something away by having a second child (undivided parental attention, being number one in the universe) I had given him something else - a sibling and hopefully, in time, a friend. Hasn't happened yet, they fight like cat and dog, but I live in hope! Your eldest will be more difficult - and harder to like - and, obviously, this is the time when he needs reassurance and to feel like his place in your affections is still secure. The wonder of a toddler's mind "I want to know Mummy still loves me so I'll be as unloveable as I possibly can."
Plus, you are exhausted from broken nights, and having to cope single-handed when your dh is away, no wonder you are feeling so down. Depression is a tricky one - I have suffered from PND and I know it was aggravated (possibly even triggered) by extreme tiredness. Sometimes it's hard to know where tiredness ends and depression begins. I would go and have a chat with your GP - anti-depressants aren't a miracle cure all but they have been a life-saver for me. They help you cope with the day to day grind a bit more easily.
It's hard I know, when you and your dh are at loggerheads all the time - is there any chance yo can go out for a meal alone, or even a quiet drink. Try and talk when you are not in the middle of it - sometimes it feels like a war zone with babies crying and toddlers demanding attention. I know that feeling of wanting to walk out and leave it all behind....
I was blessed with my Mum on my doorstep for when it all got too much, so I don't know about au pairs but I'm sure someone will have advice there for you. A friend of mine had a volunteer from Home Start when she had two children under 2 to look after. She came in once a week so my friend had a couple of hours with the eldest.
With regard to the MIL - you have to decide if the benefits outweigh the bossiness and the hassle of getting there. Could she come to you instead?
That was a bit stream of consciousness as things occured to me so I hope it's of some help. It really does get easier - but it's pretty cr*p for a while I'm afraid. Take care xx