Big post and it's late so I'll dip in on one or two....and let others throw in for the other questions.
The ONLY place you can do official translations which are notarised/attested is OFFI. You can see what it costs there on their web site.
You will need to translate your marriage certificate at least - this is regularly useful. I believe if it's in a language other than say, English or German or another easily accessible European language, you'll have to pay twice - once from say, Arabic to English, then English to Hungarian.
BTW, don't forget you need a bank account as well as electronic payments are used a lot and you will also need a card for the ATM to get HUF cash. For a bank account, you'll need your passport, possibly your rental contract and address card too plus your residence permit.
You can open accounts in major currencies without a problem - like USD, EUR, AUD or GBP etc as well as obviously HUF. I doubt you could do one in Saudi or UAE currencies but don't know for sure. You can open the account and add your husband as a signatory but he will need to be physically there. Running a bank account here is quite expensive.
To register your kids in the school, you need an address card or a contract for the apartment rental to show you are resident in the area covered by the school. Your 5 year old and 19 month old kids don't need to go to school - compulsory school starts at 6 so that might be next September depending on the child's date of birth. You can send the 5 year old voluntarily. You might be able to send the 19 month old to a play school. Depends if there is one close to your residence.
If you want to put the older kids in Hungarian school, you should get them into Hungarian language lessons asap - immediately.
Note, to go to school, the kids must be vaccinated (it's compulsory and you cannot refuse). Kids must be over a particular weight and height (that's so they can carry their school bags). You cannot get around this as they must be checked by a doctor and you need a paper from the doctor to allow them to attend - the school will ask for it.
At least when the kids are in school, the other parents will communicate with you if they can. It's quite a social system with online discussions and friendships developing amongst the parents.
BTW, up to 6, kids travel for free on the metro/transport system BKK.
Apartment rental, yes, it's safe at a notary or choose your own lawyer you trust. Do not pay cash, use only bank transfers. If at all possible try and get copies of the ID document of the person you are paying money to. Then you might have some recourse if it's a con trick.
You should really find yourself a Hungarian speaking helper to do all this. I cannot imagine you would be able to drag 4 kids around the town doing these things. Nothing is in one place.
Let us know how you get on.