Hi OP - I'm a dog walker so I hope you don't mind me giving my perspective.
Make sure whoever you pick is insured and ask to see their certificates. It's not a requirement for walkers in the UK to have insurance so not everyone will have it. But there's no excuse - you can get it from £5 a month and it covers everything from injuries to lost keys to the walkers lost income should they be out of work.
Ask about how they operate - do they do group or solo walks? Leashed or not? Do they have social media presence and reviews you can read? They should have a contact for you to sign and they should take your dogs' information in case they need to go to the vet, for example. You should have to sign and give your permission for the walker to let your dog off, and to allow them out with groups. I do a mix of solo and groups - some dogs come on one or two groups a week, some are always with a group, and some are always solo.
Ideally they will also be first aid trained and potentially DBS checked too. Again, there are no legal requirements for dog walkers so many will have nothing. You could also ask existing clients their experiences too. A good walker will answer all your questions. Also see if you can accompany them on a walk or two so you can see how they are with other dogs and how they deal with various situations.
I have keys to my clients' homes although some have keypads which is much easier! I don't have any addresses or identifying information on my keychain though, so there's no way they can be traced back to the owners.
Hope that answers a few of your questions!