2003 envoy only runs 10mph when cold when warm runs fine

That's a sign of hardened seals. They're stiff and leak when cold and soften and seal better as they warm up When did you change fluid last? You can try a fluid flush and a bottle of Lubeguard supplement. But really, start saving for a transmission rebuild.

heater core plugged. remove heater hoses, use tap water to flush out core..

You need to check the choke on the engine for cold starts it chokes off air to the fuel mix . if it is closed to much the engine will want to die and shake. When the car warms up the choke opens and the car gets air and runs fine although the choke could just be part open. It still will need adjusting.

Hope this helps

Mike

Most likely, the steering gear has an internal leak. They will usually not work well whenthey are cold, then after the engine runs for a while, the fluid warms up and the seals will expand inside the gear and start working. When you shut it off and let it cool, it will do it again. If this is what it is doing, you need to replace your steering gear.(Rack & Pinion Unit)

Rough Running When Cold GMC Yukon

You will either need to reeducate the computer or have the dealer do it for you.
If you call them they can explain the procedure.
I believe it is start the car cold, let it run till warm.
Put your foot on the brake, put it in "D" then "R" then back to park, after that it could take as long as a month for the car to finnish learning.
But I would call the local Chevy dealer to get the low down for your vehical.

that means could be vacuum leak

If you live near an Autozone, they can diagnose your code for free. This is gm's attempt to protect the engine from worse damage by limiting the power output of the motor...
it could be major, could be as minor as a gas pedal sensor...
get the code..

The thermostat is probably the culprit.....take it to your local garage and let them test the opening temp....

Have you changed the filter and fluid lately. Check your service manual to see how often that needs to be done.