You might also speak to your homeowners insurance folks and ask them if they’d be willing to help with the expense of the removal in the first place. It’s to their benefit to have a hazard removed, after all, before it becomes an actual liability to them.
As for the tree removal folks, assuming you contract with them yourself, do not just “ask to see if they have insurance”, but request proof of that in writing. Their insurer will have had to provide this proof many times – experienced companies will offer it before you even request it. You will want proof of their liability coverage – including limits of liability (in case the removal goes awry and your or a neighbor’s property is damaged) and worker’s compensation coverage (this will probably be with the state), in case one of their people should be injured during the job.
Along with the workmen’s comp coverage, if licenses are required in your jurisdiction then make sure they provide you with a current copy of that, as well. (If they’re not licensed, then they probably don’t have workmen’s comp, either.)
If the job is particularly dangerous, or looks that way to you, then it would be wise to ask for references from other homeowners in your area, so that you can check how they performed elsewhere.
See if you can sign a contract that pays them only when the job is completed – and cleaned up! – to your satisfaction. Make sure you understand up front who owns the downed timber, and if there’s salvage value in that and they take it (for pulp, lumber or firewood) that it lowers your net cost by an equitable amount.
If they’re going to have to use heavy equipment on your lawn, how will they protect that, including driving over septic tank or sewer line areas?