Here is the problem:
The car is 2000, 1.6L

The car squeak (more like a high-pitched noise) whenever I start the car cold in the morning, it happens whenever I start the car and go... IT only happens when I accelerate... and it went away after a block or two later.... It won't squeak when I idle/warm-up the car before I drive, which I usually do...

It also squeak whenever I turn on A/C before I start the car, or a couple seconds after I started the car...

The mechanic said it is related to the weak battery, lose enough juice (???) after a night at the garage... But then, I replace a brand new battery, but the squeaking problem still exist...

The alternator is making small noise, which they said the bearing inside is dry, but it is still functional... no need to replace... Before the maintenance service, there is no noise from the alternator... They did remove the alternator in order to replace the timing belt since it is blocking the way... Did they damage my alternator???

The car contains no modifications of any kind...

The car was in for maintenance service, replacing timing belt, A/C belt, P/S belt, spark plugs, wires, and water pump... All the above problems did not exist, but came out all after the service... Can someone give me a suggestion and a handy help on???

 

It could be a slipping belt. I haven't had the problem in a Protege, but my Ford Tempo used to squeak like a ****.

Since your belts were recently changed, it could be the case that they were not adjusted to the correct tension. You could take it back and ask them to check/adjust the tension on all the belts. They might not have set the tension correctly. (Or they might have. Belts can stretch a little bit, so maybe they loosened up.)

Personally, I don't see how it could be related to a weak battery or insufficient juice. Maybe there's something that I don't understand about batteries and belts, but I just don't see the connection.

One other thing you can try is to apply belt dressing. It's a sticky spray that you put on the inside of the belts to help them grab the pulleys better. This isn't a good long-term solution, because the spray wears off and you get squeaking again after a month or so. It also doesn't address the issue of improper tension. But at least it will help you determine if the noise is coming from the belts.