Actually legally you don't need to be insured for treatment liability, although in this litigious age you'd be mad not to. However you should look to get Public liability as a minimum, this IS a legal requirement ... although not so sure if it is in a home salon or if you are mobile.
Again legally speaking you don't need any qualifications, or even any training for that matter, to be a nail tech. Although of course you would be mad not to have good training if you really are serious about making this a career ... or even a good hobby for that matter! IMO nothing beats a human looking over your shoulder and seeing where you are going wrong and correcting you - a video just can't do that!