You are correct when it comes to the state vehicle inspection. You do have to have your car inspected to drive legally.
In order to have a ridesharing policy with a standard car insurance company (such as Geico or Progressive), you have to have a personal policy already established. A personal policy is what the average driver in the US currently has.
If you do not have a rideshare endorsement on your policy, they might deny the claim. Insurance companies like to know all the risks they could potentially be exposed to and ridesharing presents a big one.
Renting and borrowing a vehicle sound similar, but the way the insurance coverage works for both situations is different. Let's take a look: Renting - If you have an auto policy that includes comprehensive and collision coverage, that policy will normally extend to cover a rental vehicle as well.
As of March 20, 2017, State Farm began offering a coverage endorsement for TNC drivers in Pennsylvania. This means that you should be able to maintain your existing policy once you start driving for Uber.
Coverage & Claims
Unfortunately most insurance companies are taking a wait-and-see approach to ridesharing coverage so the answer isn't exactly clear. Each insurance company also differs in how they operate so your best bet would be to ask your existing insurance company about this specific question before you start driving for Uber.