If its only with your insurance company, generally, nothing. If you are speaking of a court action you have filed, you will have to withdraw it from the court's docket.
If you are the claimant, then you just contact the insurance company where you filed your claim again and inform them that you wish to withdraw your claim for damages. If another person is the claimant, then it is up to them to pursue the claim or not.
home inventory
You need to talk with your adjuster about the situation. The claims adjuster can allow you to withdraw the claim. I don't understand why you would file a claim, then want to withdraw it. If you have an agent, perhaps the agent can also assist you with the situation.
Ring them
As long as you have not received payment and cashed the check you can always tell your agent or the adjuster that you wish to withdraw the claim. It should not be an issue with the company unless they finished the claim and closed the claim but even then they may be agreeable to allow you to withdraw if you wish. These days you only want to make claims for larger claims. Insurance is not for small claims and it can actually do you more harm than good to make small claims. Insurance companies track amount of claims as well as frequency of claims and either can hurt your record.
No.
They should not drop you before a claim is settled. If they have contact your state department of insurance and file a complaint.
Once you become the legal owner of the home and something occurs that is covered by homeonwers insurance then it is your insurance that must cover the claim. In the event the incident is not covered by your insurance policy and you feel that a material misrepresentation was made by the seller then I can only suggest getting the advice of an attorney.
You don't need too. Just make your loss claim on your own homeowners insurance policy. If your neighbor is liable for your loss then your Insurance Company will subrogate the matter for you.
You will have to check your home owner's policy. It is a contract that you have with the insurance company and will specify how long you have to file a claim.
Short answer yes, if it is in your policy long answer, would not be smart to claim it.