Depends on what a blowout is to you. To me I've never had a blowout that was a hazard to my life. I've had some scare the crap out of me and if I had done the wrong things I might have had a spill. However just letting the bike slow down and stop on it's own with no braking or just minimal braking till I get down to a safer speed always seems to work.

You can leave the tires on the bike as long as you care to, IMO. However consider what your next rides will be. If I'm going to be doing a ride that a flat will be a big inconvenience to me or others, then I might replace those tires.

If I was going to be going fast down some long steep grades with sharp turns, I'd probably replace them now. Not so much that I'd be fearful they are going to flat. But because they look like the rubber might be hardened from age or storage conditions. Which means it won't grip as well when I really need it when going fast through a turn.