We go through Absolute Domestics and haven't had a problem with them at all. The ladies are all police checked and that's great.
Whether they bring their own stuff is dependent on the lady TBH, but they usually do all want to use your vacuum/mop/brooms etc - they may bring a product or 2, and their own gloves/rags etc. If you want them to be "environmentally friendly" then I'd supply your own products.
Getting a cleaner from an ad I'd be tempted to ask for reference checks and police checks unless they've been recommended. I've always stayed home the first couple of visits until I knew I could trust them - I'd spend time in the garden although at the moment with my lady I'm resting, and she yells at me if I get up from my rest!!!!
If it's your hubby staying at home, I can understand why they wouldn't want to be in the home with a strange man. Good luck!
Bookmarks