I'm so sorry to hear about your situation, it sounds really difficult. You've got a lot replies already, but I wanted to comment as we're in a very similar situation. I'm going through the immigration process now so I'm becoming pretty fluent in it all!!!
I'm a US citizen with British spouse and 2 children currently in UK. We're going through the green card process in order to move to the US.
Re some of your questions:
Timetable - our Petition (stage 1 of the process) took 1 year to be processed and approved (got approved May 2023). We're now getting to grips with stage 2 (NVC processing, interview at the embassy, etc) but unsure of timings at this point. A big part of the 2nd stage is the Affadavit of support and evidence of finances (showing ability to financially support your household). It is a long process from start to finish unfortunately.
Who's the sponsor - it's you as you'll be the one filing the petition (I-130 Petition for alien relevative)
Meeting the minimum income threshold - I work part-time here in UK, but my income isn't enough. So we're looking at either using a joint sponsor or using our assets (value of our house). You mentioned asking your mother but she doesn't meet the income requirement. The guidance says the joint sponsor needs to independently meet the minimum income threshold - you can't cobble together your and her income to meet it. If you use assets or savings, the amount required is also higher (x4 I believe), so it's something to think about.
See "Joint sponsor" here https://www.uscis.gov/green-card/green-card-processes-and-procedures/affidavit-of-support
Re-establishing domicile if you live abroad - I think you'll be in this position as well. Because I live in the UK but am technically the sponsor of my foreign spouse, I have to provide evidence showing I'm intending to reestablish my domicile in the US (this is why people are saying you need to move first to establish residency). However with the children this is obviously complicated as we don't want to be separated so me moving first isn't really feasible!
I'm currently working on the following to show I intend to move: renewing US driving license, setting up US bank account, getting US mobile phone, compiling my US voting records, booking one-way flights, evidencing job hunting/applying for jobs
We're also getting my children's US passports done as we haven't done that.
Other ways to evidence this are: arranging housing, registering children in local schools etc.
See here "How can a financial petitioner establish domicile?" : https://travel.state.gov/content/travel/en/us-visas/immigrate/the-immigrant-visa-process/step-1-submit-a-petition/i-864-affidavit-faqs.html#aos23
Household size - this is you, your spouse plus dependents living with you. So for our family of four the minimum income requirement to prove is $37,500.
See "How do I calculate my household size?": https://travel.state.gov/content/travel/en/us-visas/immigrate/the-immigrant-visa-process/step-1-submit-a-petition/i-864-affidavit-faqs.html#aos6
For minimum income calculator - see here: https://travel.state.gov/content/travel/en/us-visas/immigrate/the-immigrant-visa-process/step-1-submit-a-petition/affidavit-of-support.html
Sorry for the long post, but I hope that's helpful. It is very time consuming, expensive process (same with UK immigration) but I can't wait to move back. Good luck!
