I guess you're not based in the UK, but monzo(previously Mondo) (https://monzo.com/ ) sounds like the start of what you're looking for. Their API (https://monzo.com/docs/) opens them up to all sorts of uses for your bank account.
No, but we are led to believe this wont be long. They recently got partial approval with full approval expected next year. Not bad for a startup that has only been going ~20 months in quite a slow industry.
Money is currently stored with a third party that do have a license, so there are some guarantees that money wont be lost.
