I think it depends. I had someone come to clean my last house when I moved out, because I was too ill to do it myself and I needed to hand it back to the landlord in a spotless condition. They arrived in a little van with their own hoover, carpet cleaner, everything. But the few people I have known who have a cleaner for a few hours a week, every week, generally provide their own hoover (and maybe even all the cleaning products). I think if you're pushing it as a 'mobile service' etc you might need your own stuff, otherwise you just turn up and use theirs.
Yeah, I think people would be quite pernickety about every surface at least being flicked over with a duster, otherwise why pay someone to do it? At my first job in a music shop when I was 18, the boss used to run his finger round the surfaces for dust and continually complain that the poor cleaner Daisy (a plump, sweaty, sweet-natured Jamaican woman who sang hymns as she cleaned) wasn't doing her job.
You can advertise free on Yell.com (electronic yellow pages), Facebook local pages etc. Cards in newsagent/general store windows (where people place notices of things for sale).
There are lots of sites advertising nanny/carer jobs but I think you would struggle without a qualification or relevant experience. I think the days of people strolling into these jobs without either are long gone unfortunately - more's the pity, if you've ever read Love, Nina.
https://www.amazon.co.uk/Love-Nina-Despatches-Family-Life/dp/0670922765