Actually if her child really NEVER cries (not that I necessarily believe that crap!) that actually means the child is SO neglected that they have learnt there is no POINT in crying.
Years ago there was heartbreaking footage of "orphans" (they weren't really orphans they'd been dumped) in a group care facility in Romania NOT CRYING because they had all learnt they did not achieve anything by doing so, those babies are now adults and still suffering deeply from this neglect, they developed physical and mental disabilities and disorders as a result of this neglect.
I wonder if you pointed out this fact to her she would furiously backtrack? I suspect so, or do you think she is really neglecting her child this much?
NO child is "perfect"
She sounds a right fucking bitch to be honest!
Stop engaging with her, she's your husbands family not yours you certainly don't need to be dealing with daily, critical messages - that's a form of abuse in itself! No wonder you have pnd!
Frankly as someone who is very experienced in childcare and early years development myself, I see far more to criticise in her parenting than in yours!
3 months and she's regularly sending the child away for weekends?! Sorry but that is completely against all sids advice for starters and even aside from that I think that's really unhealthy.
Why has your dh not said or done something about this?! He should have! He should be protecting and defending his family - which is now you and your child above your in laws
My dd was in with us until nearly 7 months, co-sleeping initially until about 4 months and then in her cot next to our bed, only reason we moved her to her own room at that point was her dads snoring started to irritate her (it wasn't actually that bad she just sort of signalled to us she wanted away from it) she then went into her own room but often came in to co-sleep for last few hours of the night.
Ex and I split when she was a toddler, she co-slept with me again at that point as the change unsettled her, then she gradually off her own bat slept more often in her own room.
But until maybe early high school stage she'd often sleep in with me at weekends, we'd have snacks in bed and watch a movie and fall asleep snuggled up. It was me that stopped it at this point cos she's murder to sleep with and I was waking up covered in bruises on me legs! 
She never had a night away from me until she was 4 or 5? There was nobody who I could have asked, but I didn't necessarily want her to either. When she did start spending nights away from me, her father on occasion and then later sleepovers with friends, she was absolutely fine, whereas some other children the same age who we attempted to reciprocate with couldn't cope and I ended up calling their parents to come get them - I'm talking even 11/12 year olds and older.
When she started school, she happily ran in without a backwards glance, me I was a mess! 
My dd is now 20, she is one of the most well adjusted, confident, independent people of her age that I know!
Sleeping wise kid could sleep on a wire! She's slept in cars, on trains, planes, sofas, hospital chairs... it's quite a skill!
She is now away studying having moved out a couple years ago and lived on her own for just over a year but she's now staying with relatives cos she's a skint student!
You know the kids that grow up insecure? The ones that weren't supported as kids!
LOADS of research on this, seriously tons!
Being naturally affectionate WHICH IS NORMAL FOR A BABY isn't "being clingy"
You or dh - preferable dh as it's his sister and you're not well - needs to tell her to pack it in! And don't make yourself so available to her, and do some reading on narcissistic personalities and toxic families cos this sounds like CLASSIC narcissistic supply from you to her - she is literally getting off on your pain that she's creating
He’s very close with his brother so this woman isn't even his sister?! Wow! Stop putting up with crap from someone who is essentially a randomer! Dh needs to speak to his brother for his brother to tell his wife to wind her bloody neck in!
Or just tell her to fuck off! You owe her nothing, not your time or care nothing! Tell her to fuck off then block her
Frankly there's a temptation to mess with her head right back! I suspect that would be very easy to do.
NOBODY has all their shit together, nobody. Surely a couple hours on mn plus in my case 48 years of life experience teaches you that.
Strict routines in terms of what's KNOWN to be best for baby, for the most part went out in the 80's! There's one infamous "expert" who advocates routines but I really don't rate them and neither do most genuine baby development experts. I won't mention the name for obvious reasons.
her key worker commented and said she wished all of the babies at nursery were that happy and easy
Yea I call bullshit!
Not least because if a nursery worker were overheard saying something so obviously prejudiced and unprofessional they'd be in big trouble!
People who work with children don't say stuff like that in reality
Honestly, I haven’t fully told DH.
Why on Earth not?! TELL HIM tonight, the whole thing. You've done nothing wrong and you need his support with someone from his side of the family being a dick to you.