Hi OP,
Assuming you are in England, the standard time for children to start school is the September after they turn four, so your DD will start in September 2017. Children do not legally have to be receiving an education until the term after they turn five though, so you would be within your rights to delay her entry until January or Easter 2018 if you felt she wasn't ready to start school in the September. As she is born in May, you could even delay her start until September 2018, but as things stand she would have to start in Year 1 and skip Reception if you did this. A government proposal has recently stated though that it will soon be possible to 'hold back' summer born children for a year and start them in the September after they turn five in Reception if the parents feel they are not ready. Anyway, to summarise this long paragraph, the standard time for her to start will be September 2017 
The application process for her year will open in September 2016, and I believe the deadline will be in January 2017. You will have a form where you can state between 3-6 preferences (it varies depending on your LEA), and all LEAs operate an 'equal preference' system, so if you put a school as your sixth preference, but meet the admissions criteria better than someone who puts it as their first preference, you will be offered a place ahead of them. This means that you can put the schools your order of genuine preference, rather than worrying about 'taking gambles' and losing places at schools by not putting them first. It is always a good idea to include at least one school that you are pretty much guaranteed a place at though, as if the LEA are unable to offer you any of your preferences, you will be offered a place at the nearest school with places after everyone else has been allocated places, which is likely to be either a long way away, a 'bad' school, or both. You will hear rumours that you need to put X school first to stand a chance of getting in, or that the LEA has to give you a place at the school you want if you only put this school down, etc, but this is all rubbish. You should always use all of your preferences
There is no such thing as 'putting your name down' at a state school any more, except perhaps to receive information about open days, etc when the the to apply when the time comes. It certainly wouldn't give you an advantage in getting a place, as schools have to follow their published admission criteria (usually along the lines of children in care, siblings and then the distance you live from the school). If you google your LEA school admissions, there will probably be a booklet listing all schools and the distance the last child admitted last year lived from the school, to give you an idea of which schools you have a realistic chance of gaining a place at. You can list a school from another county on the form for your own LEA and they will sort it all out
I hope I've been of some help, but you may better off posting this in Primary Education, as there are experts in admissions there who know a lot more than me and can correct anything I've said that isn't right