I got stuck on No.2 on the phases.

Sorry to hear what you're going through and about your dad!

Though I'm another one that thinks get the greencard.

If nothing else, if gives you time to figure things out. If you lose status, you're gone and if the kids are here, it'll be very hard to get them back to the UK.

I got stuck on No.5. From my perspective making friends, and connections was critical. And although making friends and connections gets tougher as you get older it is not that difficult.

Another big issue is annoying the natives with negative comparisons with your home country. It is a very normal reaction, but really irritating. I was lucky that a colleague was honest enough to tell me how it was annoying everyone. And after biting my tongue I made new friends, and soon after things stopped bothering me.

Good point.
I was recently chatting to someone from another country who has lived here for a while. She didn't criticise Britain at all but when someone asked her about her home country she lit up and looked so happy, and wanted so much to talk about it. I totally related to that having been away, would not have picked up on it otherwise.

Ah UCSD. That great center of education and modern day thinking which in 2010 would only permit me to use my breast pump in the "Restroom" ie toilets.

Odd. We have a specific room for breast pumping in every building. When did you go?

Presumably, 2010...

Correct.

I was never an enrolled student but used to use the medical library loads.

If it all goes up and you end up in a child custody dispute, the legal case would be held in the jurisdiction of the kids' "habitual residence". This is not a clearly defined legal concept, i.e. it doesn't depend entirely on their immigration 'status' or country of passports, but where their 'home' (a nebulous term) is for them. If it is after living several years in the USA, I think most courts would likely find their habitual residence to be in the USA, but it's a complicated legal test to determine (you can google and find tons of information on the Hague Convention on Child Abductions)

http://en.wikipedia.org/wiki/Hague_Convention_on_the_Civil_Aspects_of_International_Child_Abduction
That just means the court hearings would be in US courts, not UK courts. You would have to convince a US judge that raising your kids in the UK, land of binge drinking soccer hooligans, away from their hard-working but ever so sweet mother, is better for the children than leaving them in the good old USA, land of awesomeness. (Exaggerations added to help illustrate the point...)

Whatever you do, do NOT consider "fleeing with the kids" back to the UK. That will be considered child abduction and the police will take custody of the kids and send them back to the USA for a court hearing. You could have immigration problems trying to re-enter (especially if there is a child abduction charge hanging over your head) and would not be able to attend the child custody hearings.

I'm going back to the original posting...
It's hard when you move anyway - it's especially hard when a parent dies at the same time. Grieving on your own is the pits and the first year is the worst. Expat marriages have a high divorce rate.
I know the OP was asking a practical question which you've all tried to help him with. My advice would be to just get through the first year, get the green card anyway, go to a good marriage guidance counsellor and try to communicate honestly and openly and fairly with your wife.
Change of job, moving home to a new country, death of a close family relative = 3 of the most stressful events in your life. If you gave up your job to move here then you can add that in too. Just sit tight and ask for help getting through it all. Get out for a beer and a laugh with friends.

if it's any help, on my first move abroad I went through all that and one more (prem baby at death's door in a NICU where nobody spoke english) It took me a long time to get over it and I wanted so badly to go home to my family but in the end I was glad I stuck it out.

also bear in mind that if YOU dont go ahead and get a green card, and your wife does, and then later applies for the childrens Gcs anyway - you may not have the right to live in the same country as them if you couldn't get her to agree you taking them to England.

so get a GC (and even better citizenship ) and keep your options only - so you can guarantee you are allowed to live in the same country as your kids.

I am in a somewhat similar position - only in my case it is my husband who has the work permit. I take the view that green card increases options for all concerned. There is nothing to prevent us all going back to the UK if we have a green card. But lack of one means we all would have to leave at the end of the work permit period - which may or may not be a time that is convenient to us.

I'm not sure I see any reason not to get the green card. Also, should you stay in the US that long, it gives your children the option to become US citizens which they might appreciate later in life.

I'm sorry you're struggling to settle here :-(