Some of the "advice" on this thread is wrong.
Let's say that on national offer day I get my 6th choice. I must accept it, right? In other words, if I don't accept it it means the council will no longer give me a place anywhere and I'll have to go private or homeschool?
You don't have to accept it but, if you don't, the council is not obliged to come up with another place for you. You should still be on the waiting list for your higher preferences, so you may get a place that way. However, if you don't, you will have to find a school with places available, go private or home school.
Let's say that I have accepted the 6th choice. Then the 5h choice becomes available. If I refuse the 5th choice, will I still be considered for my 4th, 3rd etc if places become available there?
Yes. If you are offered your sixth choice, you should automatically go on the waiting list for all your higher choices. If you are then offered your fifth choice and reject it, you will be removed from that school's waiting list but it will not affect your position with other schools.
Is the communication via the council only? Do I log into the eadmissions website, and is that where they tell me where I am on the waiting list, and where I have to accept or refuse a place? Or is communication with each school directly?
For the normal admissions round, all communication should be through the council. However, once term starts it may change, particularly for academies and VA schools. You may then need to talk to the school to find out where you are on the waiting list.