Another, often-overlook thing is that stuff - bugs, grass, gravel - can clog up your radiator and AC condenser, especially on an older car. Check and clean and you will get better cooling. If you use a pressure washer, do it with a cold engine and throw a tarp over the engine and electronics, they don't like water.

Water pump, radiator, thermostat, radiator cap, fan control module, fans, old worn out coolant, etc.  Are they all 27 years old now?

You aren't saying that you're ignoring the Temperature Gauge as it rises to Hot and then your Corolla dies.

You're saying that the engine just "shuts down" without any warning.

I'd be leaning toward an electrical issue ((surprise surprise)).

The reason I'd lean that way is because failing electrical components tend to stop working when they overheat and then begin working again when they cool off a little. (could explain that 5 minutes you have to wait before your Corolla will restart after dying in very hot weather).

"Electrical Components" covers a lot of things. Everything from a fuel pump motor to a relay to a sensor to a coil.

So in your case I'd begin my troubleshooting by checking for Spark during this 5 minute "No Start" time period after the engine dies.

If you're getting Spark during that "NO START" then it's time to check the fuel pressure for a possible issue with fuel delivery (failing fuel pump motor).

If you aren't getting Spark during the NO START then there's a couple of things you can do to narrow down the issue.

First, if you can find a helper to sit in the driver's seat and crank the engine, you can go to the exhaust pipe and smell for gasoline.

Here's the diagram that may be useful.

Notice the "Pickup Coil" in the diagram. It's inside the distributor. The computer uses those signals  from the Pickup Coil to control Spark as well as the Fuel Injectors.

The Pickup Coil is a Hall Effect Sensor and those are notorious for temporarily dying when they get hot when they're failing.

So, if you aren't getting Spark but you're smelling gasoline at the tailpipe during your NO START, then you know the computer is controlling the fuel injectors (using a pickup coil signal) so your Pickup Coil is OK. (It also tells you that Power from the Ignition Switch is providing power to the distributor so there isn't a problem in the ignition switch or one of the relays).

That would be good news because changing out the Pickup Coil in a distributor is a pain and here in the US, with a 1995 Toyota, most people would just buy an aftermarket distributor for like $90 US.

Anyway, if you refer to the diagram above, there's another component in that distributor that's notorious for having problems when it gets hot when it's failing. The ignition coil. 

It's easy to replace. You can do it without removing the distributor.

So during your NO Start, if you smell gasoline at the tailpipe and you aren't getting Spark, I'd suspect a failing ignition coil