Please don’t worry about having to apply for a place at a private school many years in advance; that may be true of schools like Eton and Harrow but is unnecessary in Cambridge.
I have extensive knowledge of private schools in Cambridge (for transparency, I live just outside Cambridge and am a parent, ex-teacher at St Faith’s and now a private tutor helping children prepare for entrance assessments to selective independent schools in Cambridgeshire).
The schools you mention all have specific assessment dates each year, which they publish on their websites (usually sometime in January/February of the same year you hope your child will start at the school in September). They also publish the deadline to register your child for the appropriate assessment (usually between October-December the year prior to entry). The exception is entry to the reception class at St Faith’s, which is a little more flexible (details on their website).
There is no need to apply any earlier as places are offered based on the school’s assessments of the children applying each year rather than time spent on a waiting list. However, all these schools are very keen for parents and prospective pupils to visit the school before being assessed. They keep a record of parents who have attended an Open Day and/or a tour of the school during normal working hours and generally take a dim view of applicants who have not visited without good reason (eg currently living overseas).
The Leys and St Faith’s are part of the same foundation. The Leys is only for children from Y7 (aged 11) onwards. St Faith's goes from Reception to the end of Y8. There is an agreement between these two schools that if pupils at St Faith’s want to go to Leys they will stay at St Faith’s until the end of Y8, then transfer in Y9. However, St Faith’s pupils still have to sit entrance tests for The Leys, and are not guaranteed a place.
The advantage of The Perse is that if your child is offered a place at The Pelican School (which goes from 3-7 years) then providing they are keeping up with their peers, (continuous assessment) they do not have to sit entrance tests for Perse Junior (Y3 - Y6) or Perse Senior (Y7 - Y13) and can go all the way through from nursery to 6th Form.
Given the criteria you mentioned, the other school I’d recommend considering is King’s Ely. I realise it’s further to travel but depending on where you live and mode of transport you use, given the ghastly traffic in Cambridge it’s often just as quick to get to Ely as it is to traverse the city! They also run minibuses from park and ride sites in Cambridge and from Y7 onwards many King’s pupils commute independently from Cambridge by train. If you drive, travelling out of the city, parking at King’s is much easier creating a more relaxed start and end to the day. Their pastoral care and support is excellent. Although children are assessed prior to entry and need to be capable of achieving decent grades at GCSE and A Level, King’s take a slightly wider range of abilities, which leads to more diversity and slightly less academic pressure. Sports facilities at King’s are amazing! For example, their boathouse on the river - King’s rowing team trains next to the Cambridge university team! Like Perse, once you have a place in the junior school, you can go right through to 6th form.
Hope this helps and good luck!