How can we persuade my father to talk to us about his health and other issues?
If you can, try to persuade your father to see his physician for a medical check-up as soon as possible. You should also talk to his physician yourself, and explain your concerns. You said your father is showing signs of mental decline. It's possible that his behavior is being caused, at least in part, by early dementia: mood swings and personality changes are symptoms of dementia.
When you speak to your father's physician, be sure to mention your concerns about his driving. Hopefully, your father will listen to his physician if it's time for him to limit his driving or stop altogether.
If your father refuses to see his physician or to heed her advice, and you don't get anywhere when you try to speak honestly with him, there isn't much more you can do to ensure his health or the viability of his business. Continue to check in on him regularly and try to keep channels of communication open, if you can.
There are steps you should take to ensure his driving safety, however. Suggest that he go for a driving evaluation from a driver rehabilitation specialist. The specialist will give you both an objective assessment of your father's driving abilities. If your father refuses to go for an evaluation, either, than it's time to file a report with the DMV.
Goodluck and let us know how it goes.