High rev after cold start

I'm guessing but this sounds like air problems. do you have a new air filter? The O2 sensor may be off but not technically bad yet. I'm pretty much stuck on something restricting the air flow.

throttle position sensor TPS is sensative to moisture.

If the idle speed is above 2000 RPM then drops down to normal, the common cause is a faulty idle air control valve or a dirty throttle body.

sounds like a failing o2 sensor. or MAF sensor. but it could also be a blocked egr valve, its common on the tdi over time. sometimes don't even register as a fault on ecu codes.

compression test
check for internal crack in turbo
replace plugs
check for blown head gasket

Sounds like a coolant sensor problem. It signals the ECU for open loop, cold start rich, high idle, until warm up is reached.

your throttle body sensor may be bad. first get a can of spray carb or fuel injection cleaner and spray it in the opening for the intake for the throttle body. this will clear out the carbon build-up. when you crank it it will run shaky but shoiuld clear up within a minute

Could be a dirty IAC (idle air control) and it's causing it to run high, also, could be a tstat going bad because it would make it run like it was cold all day...check your temp after driving for a while....see if it's lower than it should be.

could be an idle controll sensor...maf sensor or o2 sensor....run a good carb/fuel ingector cleaner through with with some high test gas, do this for you next 2 fill ups, one can of cleaner per full tank...if it doesnt work.. then one of those senors is playing tricks with ya...In worst case the timing belt is slipping..
hope this was helpful !

Here is the most common cause of surges and stalls and low idle, it is the idle speed control air-bypass valve and throttle valve (IAC for short), they get full of gunk over the miles and cause idle issues (stalls, low idle) like yours, Get a can of intake cleaner from any local parts store, not carb spray, intake cleaner, it is made by a company called CRC, remove the air intake hose to the engine, hold the idle high so the engine won't stall, then spray the can of cleaner into the intake while keeping the engine running, use at least 1/2 the can, shut down the engine and disconnect the battery for 5 minutes, then restart and complete a number of mixed driving cycles, town, freeway, stop and go etc., after a few days the problem will go away as the system will relearn to the clean intake