My brother's car has a strange proble.

Here's his description:

Driving around town speeds, the engine will start running rough, the low
coolant level light comes on, and the check engine light comes on also.

He also said at that time the brakes act funny and the transmission
shifts hard. This occurs randomly, about 5-10 minutes after startup.
Sometimes it will go for a week without acting up. He took it in to a
shop, and they diagnosed(?) a bad TPS. They changed it, at the cost of

over and tried to check it out. I thought thermostat/water pump. I
changed the thermostat, took off the serpentine belt and spun the water
pump by hand. It wasn't weeping out the hole, and no rattles or
looseness. I put the belt back on, filled it until the bleeder above the
water pump started dripping, and started it up. I bled the air out of
the bleeder above the water pump and closed it. Before the gage even got
to normal, and the thermostat could open, the car died. At the time it
died, the radiator fan(s?) came on. I restarted it, and it ran really
rough, then the engine idled up to about 2 grand for a second or two,
and idled down to normal speed (and smoothness). The thermostat opened
after a little while and I closed the bleeder above the thermostat. I
then topped off the radiator, shut it down, and filled the coolant
reservoir to the proper level.

I have a few questions.

1) Why did the fan(s?) kick on when it died, when it wasn't even hot
enough to open the thermostat?

2) Why the roughness and fast idle for a few seconds?

3) Any ideas why it's acting like this?

He bought the car from a friend's mother, and he got a stack of regular
maintenance records, and all repairs that were done at the GM dealer.

I looked through them and found no abnormal repair receipts, just an EGR
valve, alternator, and a front wheel bearing, other than normal wear
items. The car has 93k miles.

It truly is a grandma car, excellent interior, body, etc.