As an ex mobile hairdresser I would say it's not nearly as flexible as you would think (sorry!) as with all self employed trades of course you don't answer to a manager or a boss but you do have to make money and most people will want an after work or weekend appointment - I almost never got any bookings before 2pm. I did it for four years, there were some great perks but I wouldn't say it was family friendly, unless you stipulate you only work between 9 and 4 (or whatever) and even then I think you'll struggle to get a booking a day (which was always my minimum, to be able to cover petrol, insurance, materials and £6.50 an hour for myself to keep). My busiest time was between 4 and 9pm, sometimes I didn't get home until 10:30/11pm which was okay for me as a single person but if I'd wanted to stick at it once being a parent I would've had to have gone back to working in salons and had super stellar childcare which just wasn't my thing, but I was always home by 8pm when I did salon work (6pm "close" but appointments would generally run over to 6:30 or 7;00).
So your degree of flexibility will depend hugely on your target market and the area you're working in, but IME those who want earlier appointments will just go to a salon.
There is a lot of suspicion towards mobile hairdressers as it's a highly unregulated industry with a lot of cowboys - same goes for salons but it's a lot harder to buy a salon and call yourself a hairdresser with no training than it is to buy some scissors and nice deep conditioner and call yourself a hairdresser. So a lot of people can be suspicious and won't take a chance on a mobile hairdresser unless necessary (ie, kids at home to watch or they need a 7pm appointment etc etc)
It's a great job, very social, very active And you can make great money but unless you're willing to work for very little and don't take many bookings I wouldn't say it was particularly family friendly.
I can't help much with training costs as mine was via full time college and as such costs were minimal (£250 a year, I think, for 3 years, split into a term payment plan) but make sure that you go with a well regarded and accredited course. Some places claim to be able to train you in a day for £750, and while they can give you a certificate that's not worth the paper it's written on, they can't actually provide you with proper training in that time (see what I mean about it being a highly unregulated industry?!)
Sorry if I've pissed on your chips!