No, they won't have insurance, and any injury they suffer on your premises could be charged against your insurance.
Any damage they may do to your property equally may make them personally liable, but good luck recouping that, so again you'll have to claim from your insurance.
In terms of the type of work they do and whether that will invalidate your insurance, the rough rule of thumb is that if it's something you shouldn't be doing, then a "handyman" shouldn't be doing it either.
So simple thing like installing a dishwasher, hanging shelves, hanging a TV, wiring up a network, hanging a door, all fine. Changing a ceiling light or adding sockets are things I would only do myself or pay a sparks to do, but again will be covered by your insurance.
Any work on gas systems, plumbing systems, larger electrical works or structural works on the house itself, will not be covered by your insurance unless carried out and certified by registered professionals.