Glaciferous is broadly correct. To give a little more detail:
Does the school in your catchment area have to take your children if you move there?
Most schools in England don't have formal catchment areas. But the short answer is no, your nearest/catchment school doesn't have to take your children. If it has places available it must offer you a place when you apply but if it is already full you won't be offered a place. If you are not offered a place you can appeal. Your chances of success depend on whether or not it is an infant class size case. If your child would go into Reception, Y1 or Y2 there is a good chance it would be an infant class size case which means an appeal would be unlikely to succeed.
What if it's full?
In some areas all in-year admissions (which is what your child would be) are handled by the council. That means you would apply to the council naming a number of schools as preferences. If none of your preferred schools have places you will be offered a place at the nearest school with places available. If your council requires you to apply direct to schools they will only step in when you have failed to find a place. Again, you will be offered the nearest school with places available. However, if the nearest school is an unreasonable distance away (more than 45 minutes each way) the council should use its Fair Access Protocol to force a school nearer you to admit your child. Note that you won't get any choice of school in this situation.
Do primaries feed into specific high schools, so should I look at the High School league tables and work backwards from there?
Mostly no, but some secondary schools do name specific primary as feeder schools. You need to check the admission criteria of the secondary schools where you will be living.
What if we rent first in one catchment area, get our children into the school and then move out of the catchment?
It depends on whether the council think you have rented specifically to get a place at your preferred school. If you apply to a school with a place available they must offer you that place regardless of where you live so there is no problem in that situation. If, however, the school does not have a place available, there is a waiting list and you get to the head of the waiting list because of your rented property the council may well think you have fiddled the system and remove the place when you move away.
our eldest having a diagnosis of HF ASD
It is unlikely that this would give you any priority. Even if it did, it would only put him into a higher position on the waiting list. It wouldn't result in him getting a place immediately. For this to happen you would need a school that has "special medical needs" or similar as one of its admission categories and you would need to show that this is the only school in the area that can meet your son's needs. Since all schools should, in theory at least, be able to cope with HF ASD this would be difficult.