A good underwriter would be looking at the stove to make sure that it was installed with the proper clearances, according to manufacturer specs and any local codes and to make sure the customer is safe in handling the ashes. While an underwriter is a good generalist, they won't know all the rules that are applicable to each and every stove in each jurisdiction. So they will ask whether it was installed by a professional as a proxy for all of that - since any professional should be taking all of those things into account. Worse case, if you got a fire due to a botched job, the company could subrogate (sue the installer for the amount of the claim they paid to you). With a homeowner install, they have to do a little more work to make sure its safe. They won't want to just take your word for it that you know what you are doing. They will be able to tell you story after story of someone who burned their house down doing something they thought they were qualified to do and weren't.
Its true that a CLUE report contains claims from the prior owner of the house. It may cause some stumbling blocks when buying the house and setting up the policy in the beggining. They may want a more intrusive inspection or have you promise to get things fixed or inspected. Its really uncommon for another person's claims to impact how much you pay. The claims follow the person, not the house in pricing decisions. Companies usually have to disclose claims that have a negative impact if they got that from a third party like a CLUE report to the customer. If you find you are paying for the sins of the prior owner - shop around.
If you want to save money on your insurance, the best thing you can do is select higher deductibles for covering your property and high limits when covering liability for others, don't use your home policy like its health insurance to get reimbursed for frequent and small claims that are really just maintenance. If you make updates to the house, let them know - you may get a better rate, especially for things like updating your roof. Also keep an eye on your credit score. Like it or not, a credit score is a report card for grown-ups and as good a predictor of future losses as anything else and almost all insurance companies will use it in their pricing.
Sorry for being long-winded. This is about the only thread on here where I actually know what I'm talking about and am just trying to give back for everything I've learned here from all of you.