Do I have to pay the tree guy for his services if he dropped the tree on my house and his rope broke and tree fell wrong way?

He pays for what damage he causes.

Yes, and he has to pay for the damages he caused, call you homeowners insurance and make a claim for the damages.

Generally you would pay for someone's services; however, if they negligently provided those services it's probably best to discuss the situation with them and come to some mutual agreement about the bill/reimbursement for damages. If you can't reach some mutual agreement, you could not pay the bill and wait and see if he sues you, at which time you'd answer the suit and file a counter-claim for your damages (unless you had your insurer pay them). You could also either pay, or not pay, the bill and sue him for your damages, in which case he'll likely counter-claim for the amount of the bill. You could turn over your damage claim to his insurer and/or your insurer, or both, and see what happens before deciding if a suit is needed, which is the way I'd suggest handling this if you don't reach a mutual agreement. Remember any litigation takes time and money and unless the damages warrant same, litigation may not be either time or cost effective. Thus I would advocate for you and the tree fellow to reach a mutual agreement without litigation.

You do have to pay for the service, but he has to pay for the repairs to the house.

You didn't hire him to negligently drop a tree on your house, so, no, don't pay him, and instead make demand upon him for the cost to repair your house. He in turn should turn your claim in to his insurance company.

No make him pay for the damage to your house.

Well, he has to pay you for the damage he caused.

Technically speaking, I suppose you do, you hired him to remove the tree and he removed it. But of course, he has to make good on his damages. Your homeowners insurance company will pay to have your home fixed and then they will go after his insurance company for reimbursement, plus your deductable. Meanwhile, if you don't pay him, he'll probably be too embarrassed to sue.

Probably not. Contracts to perform services in Ohio include an implied warranty of workmanlike service. Absent language in your contract to the contrary, the tree guy probably violated his warranty. You should be entitled to offset any amount owed under the contract against the damages that you sustained from the tree guy's breach of his warranty of workmanlike service.

If he had liability insurance that is paying to fix your house, then yes. Otherwise, no, and it sounds to me like he's going to owe you a LOT of money.