2008 Impala squealing noise on start up

serpentine belt is getting old and hard and the cold weather was causing it to slip
check belt condition ( small cracks across the grooves = scrap and get a new one) and check tension

Chevy Impala 2000 2005 Factory Service Workshop repair manual

for the easy solution download the pdf file

What is causing that irritating noise? ' DSMtuners

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Try to tighten the belts a little firm.Sounds like when engine idling no squelling only when about to take off .Some times belts may seem ok on the outside but if you look on inner side you tend to see cracks in the belts .

Could be several things including a bad tire. The computer counts the number of times each tire turns and compares that number with the other tires to make sure none of the tires are slipping - skidding.
You should be able to get the body computer scanned by a shop and the trouble code will tell them what to test.

It might be the actuator on the ac/hearty door making tapping noise

sounds like the clutch on the compressor

There is bulletin for a humming noise like yours that has to do with the Torque Converter. #01-07-30-010C 5/12/2008 Call GM and ask about it.

Check your drive belt tension.

Sounds like the new alternator was run without first fully charging the battery (12.3 - 12.8 volts)
This will destroy an alternator every time -- the classic symptom is a squealing belt.