There are two ways you can win an appeal for a secondary school place.
You can win by showing that a mistake has been made and your daughter should have been admitted to the appeal school, e.g. she has been placed in the wrong admissions category or the home to school distance is wrong. It does not sound from your posts like you have any evidence of a mistake. However, if you do have evidence, this is the strongest kind of appeal.
If no mistake has been made, the only way to win is on what is known as the balance of prejudice. This means you have to show that the disadvantage to your daughter from not being admitted outweighs any problems the school will face from having to cope with an additional pupil. The disadvantage could be subjects or extra-curricular activities the school offers that she will not be able to access without being admitted and that are particularly relevant to her, for example.
The panel will be looking for things that disadvantage your daughter, not things that are a problem for you. For example, they won't be interested in transport or childcare issues - that is regarded as a problem for you, and children of secondary school age are generally able to get themselves to and from school without needing parental supervision.
Remember that you are appealing for the school you want, not against the school you've been offered. Arguments about the allocated school not aligning with your values are unlikely to fly, and the panel may well disagree with your view that the journey to and from the allocated school is unsafe. I would also avoid talking about an "LA dictatorship". That won't help you and it may make the panel less inclined to give you the benefit of any doubt.
As others have said, you need to be ready for the possibility that you will lose all your appeals and have a plan B. You may get another offer through the LA, but that is not guaranteed. They have already fulfilled their duty by offering the place you rejected, so you will probably only get an offer if a place comes up via the waiting list. If they don't, you will have to either home educate or send your daughter to an independent school. Even if you are happy with those alternatives, I would have recommended waiting to reject the offered place until your appeals had been heard. The fact you have rejected the place offered won't help you at appeal and could make it harder for you to win.