1. Contract
Yes, you need to negotiate with the company for the deal that you will be happy with and that will be of benefit to you in the long run, so that includes education allowances, housing etc
If they are paying 21K per child in education then you would be required to make up the difference if there was a shortfall.
My advice is to ask many, many questions to clarify and "nail down" what is/isnt included, as it is common that the wording may be interpreted in a number of ways and you need the clarification in writing to support any claims you have later; for example do they explicitly state that amount is per child or are you assuming?
2. Schools:
American school is at the upper end of the market. Other schools are cheaper (30-40K/year secondary), but as I work in one I can say the staff are great and ready to teach, but there are often behaviour issues with local Arab students that also attend, which cause distractions in class and impact on learning, although we try our best to limit it! If your kids are focussed, motivated and bright OR you are the type of parents that will closely monitor what is going on, they might be OK. Personally, I know my kids and I would not have them in the secondary school where I teach as they would not perform to the best of their ability.
Unfortunately, school places are highly sought after and as scarce as hen's teeth! So, you may have difficulty getting a position in the first place and then the place you get might be in one of those 'lower' schools - its the "rock or a hard place" option if that is the case! Your other option may be to homeschool, which there are many who do - Qatar has many clubs and if you are on a compound then the kids would still have social interaction, just not in "school".
3. Housing
It could be on a compound or it could be in one of the thousand apartment blocks scattered around Doha - this is another area I would seek a great deal of clarification on - such as where it is located, could you have photographs of what it looks like, what other facilities the apartment block has etc..... it may also be wise to negotiate an alternative - that if you find this accommodation is unsuitable that you have the option of taking an allowance so you can find your own place. A 3bd unit/apartment unfurnished anywhere in the city will set you back between 13-15K per month in rent atm and 2bd are often in West Bay (city) or The Pearl and start from around 15K per month or more.
4. Sorry I can't help much with the University question - but yes I would think you would have to pay - however I can let you know some of Universities that have campuses here if you need that information.
5. Visa
Check the Qatar website - you may be able to bring your family with you from the moment you arrive if they hold passports that are eligible for "on arrival tourist visas". These last for 30 days and a short trip to UAE/Bahrain is all that is required for that to be renewed - although in some cases you can ask for and be granted an extension.
You will arrive on a business work visa and the company will (at some stage) begin to process paperwork for your Residence Permit- (another lot of questions to ask - like when will this start - will it be before or after probation period etc). This is important because without it you are not able to buy a car, open a bank account, obtain a local credit card, have a licence (for car or alcohol) and many companies will not rent a house to you. This process can be quick or long - but I hear the 'average' time is 3mths.
Once you have your RP you can then apply for the family visa - this may then take several weeks to process.