IT starts @ the intake manifold & ends on the front axle.
If this happened all at once, check the vacuum lines. A cracked or broken line will cause high or rough idle.
The 02 Chevy S10 with a 4 cylinder or a 6 cylinder does not have a timing belt.
You failed to mention what year the S10 is but, most of them use a fusible link to protect the alternator circuit.
Considering the Silverado has a 6 lug rim and the S10 has a 5 lug rim, it would be an expensive project to convert the 5 lug S10 into the 6 lug pattern of the Silverado.
Test it with a vacuum gauge.
GOOGLE!
Look under the battery tray.
check for a vacuum leak.
It doesn't have or need a line connected to it. The spider is inside the vacuum chamber.
you may have a vacuum leak somewhere.
http://ui.abimg.net/images/answers/598789/2282485/1988.S10.Blazer.Vac.jpg?1328637075a
Some of the HVAC controls are vacuum operated so, yes a vacuum leak could effect the cooling.
on the top of the transmission its on the transfer case with 3 vacuum lines coming out of it.
A plugged catalytic converter could do that.
Plugged/damaged catalytic converter.
look under the battery tray