My kids don't do any peri lessons, so it is not necessary. We have never been made to feel strange for it.
If you want your child to learn an instrument, it's not a cost solely to do with private schools. If they learn an instrument at state school, you will still have to pay.
Ditto uniform - may be a bit more expensive, but generally good quality and they really do take care of their uniforsm. Shoes - you have the same range that state pupils have.
Educational visits aren't subsidised, so can easily be £20 - 30 a go (museum trips, theatre, field trips), but you would unlikely have more than 4 or of these a year.
Lunches - they still have to eat no matter what school they are at. Some schools have the lunches included in the fees across the school, or compulsory for younger age groups, with half-termly notices for the other age groups. Like for like, school lunches shouldn't be a worse deal than in state schools.
As for fee hikes - it depends and is usually decided at Easter each year. They look at the number of pupils, and the outgoings, and work out the fees accordingly. Sometimes fees can go up a lot (10%) and sometimes a little (3%).
The main cost is that of the teaching staff, and the school has to fund the annual teachers' pay award set by the government. They may have additional teaching costs as teachers gain experience, but may save when an expensive teacher retires and is replaced by a NQT.
There are other costs that need to be passed on, such as insets and teaching materials to cover all the new GCSE and GCE courses, and the new culture of providing text books for keeps now that they contain CDroms (licensing issues).