I strongly recommend insurance, both for your business assets (those physical things you use in your business) and liability coverage (what if someone were bitten by your dog or slipped on a rug?). With more and more businesses in the home, the insurance industry has had to address this problem and create products for this market.
The first step is to call your agent and find out if your homeowners'/renters' policy covers a home music studio. (If you rent commercial property, you will have to buy a commercial policy.)
You need medical as well as bodily injury coverage. See your agent.
Consider getting a "zero-deductible" rider on these items. The difference in premiums is not that great, and your belongings will be replaced without your having to pay a deductible.
Also make sure you get a zero-deductible policy for "replacement value," not "depreciated value;" otherwise, you will get only a portion of their value returned to you. A zero-deductible policy for replacement value means that your teaching equipment is replaced immediately and in kind and with no further out-of-pocket expense.
When talking to your agent, give her a couple of scenarios and make sure the answers are what you expect. Ask, ask, ask.