Hi I ived in Riyadh for two years, my husband is still there. Your husband will go first when his paperwork is sorted, then when he has his iqama (residence permit) he can start to apply for yours and the childrens. You will need to have a health check invovling a blood test and a chest x-ray - I got mine done in Harley Street at a clinic that specialised in this. You have to apply for Saudi residency in your country of birth - not sure this is relevant, but we had to travel back to UK for a week to get mine done). When your paperwork has been completed you will be able to travel.
Re accommodation - it is very hard to get accommodation on the better Western compounds - if you search you can get details and contact numbers and I suggest you start calling them as soon as the job is confirmed. You can then get on the waiting lists. The better ones are Kingdom Compound, Al Hamra (which is connected directly to BISR), Arizona (has a golf course), Cordoba.
Re BISR - my two went there - it wasn't great tbh, but has had another change of head recently so may have improved. They had problems keeping head masters for a while, not sure how things are now, but I have a friend who works there so will ask her and get back to you.
It certainly isnt the easiest place in the world to live, but if you go in with an open mind, knowing the restrictions it is doable. You won't be able to drive, but the compounds do shopping buses that go weekly to various malls or you can hire a full time driver to take you to whereever you want to go, the compounds also have drivers that you can use on an ad hoc basis. You will have to cover up (abbaya) but I nor my daughter ever covered our heads despite her being very blonde. We kept our eyes open for the muttawa and avoided them.
You can get most foods there now.
I worked while there which saved my sanity - if you are a coffee morning type of person you will be fine, but you do need a hobby or interest or job to keep you going as there is very little to do. If you are sporty that is good - the compounds have pools, gyms, basketball courts, golf etc. You can horse ride but it's about an ours drive from Riiyadh.
My kids loved it, I loved my job. My best friends were Lebanese - I learnt an awful lot about tolerance during my time there, and also that I don;t have to put up with idiots,
Hoe this help and I'm not teaching my grandmother to suck eggs. PM me if you want to chat.