Yes, you can be covered to drive the vehicle if you are listed as a driver on the insurance policy for the vehicle. Typically, car insurance companies will allow you to be added to the policy with an expired license if you get your license back within 30 days.
Your rate may be impacted by the prior suspension if it was within 3 years of the start date of a new policy, but each company is different so there is also a good chance that it won't have any impact. If you speak with an agent in person or over the phone you may want to mention the suspension but also note that it is currently valid to avoid any confusion since the driving report is run at the time that you purchase your policy and not when you get a quote. I'd recommend quoting with as many companies as possible to make sure you are getting the best rate.