Are osteopathic doctors covered the same ways mds are by my insurance plan?

Usually yes, If they are providers under your insurance plan. Most dos accept the same insurance reimbursement as mds, but check with your individual doctor, regardless which degree they have.
Yes. Osteopathic physicians are licensed to practice medicine the same way that md's are licensed and have the same coverage by insurances just as any other physician. However, most do's under insurance plans are usually board certified and i would look to this as a qualification as well.