First of all a big for you, your doing an amazing job even if you don't think you are.

1. If you co-sleep in your bed does your partner wake a lot or your just worried the noises WILL wake him?
My DP works long hours and at the start we had DD2 in our bed a lot, i was worried she would keep him awake but after a couple of nights he slept through her cries and it was no longer an issue, she was comfy, i was comfy and we all got some much needed rest.

2. Could she have a bit of reflux? Have you tried propping her up a little when you put her down, she might not like laying flat and thats why she enjoys being cuddled?

Do you have a carrier? DD2 is a velcro baby, was permanently attached to me when she was younger and would scream if i left the room, i found using my carrier worked wonders, she would sleep if she needed to otherwise would just look around and enjoy mummy cuddles.

As to how do we do it, i don't know, there are days where i honestly think OMG why on earth did i have these kids, DD1 like to 'help' with the washing and stuff now so i just move DD2 from room to room with me and give DD1 a cloth to pretend to clean, it's the only way for me.
everyone has their own way of doing it, you WILL find what works for you, then just go with it.

Oh and a big for your partner looking after her when he gets home.. i could only DREAM my DP would do that, i never got that luxury, he came home i was cooking tea and juggling 2 kids... maybe you can compromise, he takes her for say an hr while you nap then he can study.. would that work?