General

yes you can drive by yourself with that, that's why they give you that paper until you get you're real license so you can drive right away

and yea you need to be added to you're parents insurance policy if you're going to drive any of the cars in your household

Interim license as in that piece of paper they give you until your actual license arrives? Yes you can drive by yourself with that, but you have to have it in the car with you at all times. However, if you're under 18 your license will be provisional until your 18th birthday, and that prevents you from driving around with other passengers unless there's a licensed of age driver present, and you wouldn't be able to drive after a certain time of night. But if you're already of age then that's not an issue.

And the car itself definitely has to be insured, but if your parent's insurance covers you (or at the very least covers incidental drivers), then it counts for driving the vehicle. However, it doesn't hurt to be sure and double check on their policy, because as far as car accidents go, if you aren't covered under their insurance you're basically the same as an uninsured driver. But I don't think you can get cited if you at least have proof that the car has some sort of coverage.

I live in CA, but I've had my license since late 2005, so I haven't had to check the drivers handbook in years, so some of my info may or may not be outdated lol.

Well of course you have to have insurance when you drive thats pretty basic and all your answers should be in the drivers book that you had to read through in order to pass the written test so consult that for you state policies on this matter

i don't think anyone outside of your state can answer your question
but for my province (canada), there were two tests- written and road test; pass written test and i can drive with another person in the car with proper license and pass road test to drive by myself without restriction