Answer to both your questions: Most likely.
There’s no real way to determine if 5 year old tires will fail before they reach the end of their usable tread life. There are many factors to consider, including temperatures, moisture and how hard you drive it. We cannot guess as to how long the internal making of the tires will last. I’ve had brand new tires slip belts, and old tires work just fine. It’s a coin toss, really.
If they appear to be in really good condition, and you still feel safe on them, and you have a current AAA membership (just in case), then consider keeping them.
Having said all that, is 5000 miles worth the cost to your conscience? Replacing them now will ease your mind, and you most likely still have time to save some money and buy them in a few weeks.
Was your mechanic willing to put that statement of his in writing? If so, he’s guaranteeing you they’re OK…but I doubt he did.
Good luck!
Chase