Good evening, OP!
Congratulations on the birth of your son, so great that he is out safe and sound. I hope your partner is able to rest a little despite the whirlwind that is having a newborn.
I think the two issues you describe are very, very common, and definitely to be expected given your son's dramatic arrival. He has spent 9 blissful months safely cossetted in the soft, rosy glow of your partner's womb only to arrive in a jarring new environment after a stressful journey. Birth is rough on babies! (And on both parents 
)
Breastfeeding can take a while to establish. There is lots of information available and people will be along here with lots of advice, I'm sure. I found that time was a crucial factor; no expectation on me to do anything other than establish breastfeeding for as long as it took, so just staying in bed, snuggling with baby skin to skin, letting baby sleep and offering a nipple every so often, for hours on end, for the first week (both my DC first fed just lying flat in the bed, 'finding' the nipple when offered). The housework put indefinitely on hold, knowing that once breast feeding is established, everything can slowly get back to some new kind of normal. And if it doesn't work, there is always expressing and bottle feeding or using formula. Get your health visitor to check for tongue-tie and support your partner.
As far as the sleep is concerned, I think it is totally natural for your son to want to sleep in close physical contact with you or your partner. After all, he has been physically attached to another person for the whole of his life so far, and this is what he knows and feels comfortable with. I'd say let him, and ignore the 'you're making a rod for your own back' people. Being close to you helps his tiny infant system regulate in a new and overwhelming world. He is still so very new, and his system will still be flooded with all the stress hormones from birth; keep him close, minimum stimulation, still, quiet, low light and let him arrive in this new reality gently. Let him sleep the stresses of birth off, snuggled close to you; get comfy with a good book or movie and just let him sleep. And sleep. To be honest, this is more important at this stage than putting him down and kick-starting 'good habits' -there is loads of time for that later on. You'll be knackered, uncomfortable and wondering when he'll learn to settle in a bed, but just let him sleep however he needs it, and if that is on you, then so be it. He needs to sleep. Tag team with your partner to have him sleeping on your chest, pop him in a sling suitable for newborns (inward facing), and keep him close. Chances are that once he has rested up and his system has had a chance to regulate and settle, he will be more able to feed too.
Birth is a life-changing time for everyone involved, especially if it is your first baby. Nothing prepares you for the surrender of your own routine and preferences, the patience required is nothing short of saintly. You'll likely be tired beyond belief, and wonder at how slowly time can creep by in the early hours of the morning. Be kind to yourself and your partner; you have both been through something huge. She is likely to be really sore, if not in outright agony, so do everything you can to make life easier for her, even if she is putting on a brave face and making out she's OK. The brave faces of new parents are very brave indeed. Do whatever you can, whatever it takes, to keep things light and sweet at this time, and you'll find it is all so much easier. And your boy, he is relying on you both to provide regulation (babies can not self regulate or self-soothe), so give him what he needs (he's showing you; he is contented when he has what he needs).
Best of luck, and welcome to Mumsnet, where you and your partner will find endless advice and support from other users. I reckon there must be a thread in the archives on almost every aspect of parenting, however obscure.