In statement analysis everything the subject says is important.

In statement analysis everything the subject says is important  whether it be vague, I got up went to work  went home etc or if it goes into minutiae, I got up i went to the bathroom, I washed my hands etc.

When we talk or write we self edit, erasing the pointless trivia we do each day and talking about what is actually relevant to what is being discussed.

When we analyze we look for the important things in relation to the subject not ourselves.

We look for pronouns, where the subject takes ownership of what is being said (I) we look for where they go missing or change and if the change is warranted.

We look for omissions in time (temporal lacuna) where they occur and if the gap is relevant or not (gap in time during a crime or event is important, gap in time going to sleep and waking up is expected)

We look at the order of things, order is important, where does the subject start their statement (this is important to the subject) what order does he place things (family, friends, workmates etc, the order tells us what is important to the subject, if he says family, friends, wife then it is probable things aren't great with the marriage)
We look for chronological order, an event will follow a certain path, beginning, middle, end, if the subject jumps around say middle beginning end or things are out of chronological order (i gave him CPR, the shock hit him and he collapsed, he wasn't breathing i couldn't feel a pulse. You get the drift)
We look for the expected  thus the unexpected will stand out.
We assume the subject is telling the truth thus anything  unusual or unexpected will stand out.
We know fairly well what we would do in a certain scenario (in my case  see a spider, I watch it closely whilst I grab the nearest spider eating cat if it is within reach or they are around, if they aren't,  depending on location it is the slipper, if it is big and high it is the Dyson of doom, the fastest and scariest ride the spider will have.)
We know to a great extent how things will go in order of priority, family, friends etc.
With children we usually go in order of age, this can change depending on topic to boys/girls, girls/boys, trouble/not in trouble, talent/no talent.
We know thus anything out of order will catch our attention, it may be something innocuous and relevant only to the subject and will make sense in context.
It may be something  incriminating, say a man with a missing wife will put a close female friend before family, friends etc. it gets our attention because it is unexpected, we would put family as more important than friends or an individual.
She isn't a family member, is a friend, yet warrants priority over both, she is important to him , enough that he gives her priority.
Later we see he killed his wife and was having an affair with the woman.

What is interesting in gerry's blogs are the priorities.

On first publishing, on the page that was supposed to be all about Maddie, we would expect to hear about her character, her likes and dislikes, a description, things that were important to us to help find her, a description to humanize her, describe her as a little girl needing to be back with her family rather than a body.

She warranted a whole 7 words which was entirely unexpected and a huge red flag to me.

The rest of it was about the twins, gerry and kate.

It was as if she was nothing to the family, a pet, an object rather than a living breathing, loving little girl.

Obviously gerry was paying attention to blogs as I and many others commented on this glaring red flag and subsequently amended it.

Gerry is very aware of what is said about him and will change things where possible to paint him in a good light.
Where he can't enable a change, he will minimize, redirect attention to someone else or sue or threaten to sue to have the comments removed.
Classic narcissistic behavior.
I suspect at home as the baby he was spoiled and as he grew up blamed everyone else and their dog when he was naughty or something didn't get done.

What I note in the blogs is how little is about Maddie and how much is about him, the twins and kate.
How he goes into minute detail about irrelevant stuff, how anyone helping out is minimized or downplayed whilst he does the great I am.
I did all these meetings, spoke to all these people, oh and family members cooked dinner after a really long day.

I wonder if he even wanted kids?
Kids are selfish little buggers, gerry is a big selfish bugger.
It would have been chaos in the house with 3 small toddlers and a big one all making demands, it is no wonder kate obsessed about sleep.
Kids would be competition for  kate's attention, and competition against gerry.
I wonder if kate wanted children and gerry didn't?
When they found out they couldn't conceive naturally, was gerry relieved I ask?

Who pushed for IVF?

I wonder who was at fault for the infertility?
Kate would feel guilt (not a perfect woman, catholic doctrine even if not a devout one it would be in her psyche)
If gerry then would he feel anger rather than guilt (huge blow to ego that he wasn't a man's man, he wasn't perfect, he who demanded perfection from everyone and everything, who always got his own way couldn't father children naturally)

IVF is hard on the woman physically and emotionally, hugely expensive especially with multiple cycles, stressful waiting for a positive on the test.
How many cycles did she go through before each success, what type of IVF did they have?
Were donor eggs or sperm used?
Are the parents of the twins the same parents of Maddie?

There are various combinations of egg and sperm from kate and gerry and donors.
I would be interested to know what type of IVF were used and who provided what since it could provide a motive for one and explain the distance between kate and gerry and the twins.

Even if both were the parents, the fact that IVF was used could be a motive, there was no 'bond' so to speak, and, given that she was born almost perfect, we have motive for her death and also an explanation for their distancing from her.

She wasn't perfect, she was conceived in a test tube so not 'really ours' she had health issues (the eye) behavioral issues (temper, demanding attention, screaming possibly autism spectrum perhaps?) 
Imperfect was not acceptable especially to gerry. 
Maddie demanded and got attention whilst gerry didn't.
Over time this can build up resentment, anger and ...

The sedation was perhaps kate's only way to maintain her sanity.
A sleeping child is a safe child, she can concentrate on gerry and a happy gerry means a safe kate.
 A safe kate is a happy kate.

We know he chased her all the way to NZ to woo her, I wonder if perhaps she went to the other side of the world to escape him?

These are all suppositions of mine, my own opinion on the family dynamics.

What is clear is that this is a family in a toxic relationship, kate has warned us of her possible intentions. 
It could be said she meant nothing by them, this is wrong.
She meant what she said otherwise she would not have said it.

The Freudian slips where they talk about murder and death, these are what is at the front of their minds when they are talking.
The mind thinks the words a microsecond before before spoken or written, it is what they are thinking of when they speak.
It is the brain speaking the truth despite them trying to prevent it doing so and self editing.

Lying is stressful and the body doesn't like it thus the brain will do it's best to relieve the stress by telling the truth.

Frequently when a suspect in a crime finally admits the truth, they speak of the immense relief, the load has been lifted from their shoulders, they feel so much better.

Listen to what they say, read the words they write, don't assume they meant something else or it was a misspeak, it isn't and wasn't, it is the truth leaking out.