Makes squealing noise

when changing a serp belt i recommend changing tensioner and idler pulley's also that could be causing the squealing you mention were all pulley's checked for free spin when belt was off ? Not really sure what would cause hesitation unless you have a serious drag on serpentine belt

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 .

As I look at your letter, which is very good, I might add, here are the few things that occurred to me.
The "loud squealing" should be a belt.
You have one or several belts, and if they get loose, or are worn out ( that includes the belt rubber getting old and hard) they will squeal like that.
However, the squealing also may be coming from one of the pulleys the drive belt or belts are attempting to turn.
If you have a bad bearing in one of the pulleys, you can get that loud sound like " a train".
If you are really careful, a "mechanic's stethoscope" might help you.
That is to take a really long screwdriver, and put the tip on each of the pumps or base of each pulley, and the handle end to your ear, and you can hear the bad bearing "scream" with the engine running.
Be blessed.

Squealing is most common with loose or worn belts, I would start by replacing the drive belt on the front of the motor and go from there, Its easier to troubleshoot a squealing motor if you know that the belt or belts are good.

The squeal is coming from the alternator belt, When the temp is low, the belt is hard and glazed and lost its ability to create friction.

If you get the squealing noise even when car is not running, and if noise get louder when you push the accelerator pedal, then it is the serpentine belt.

You may need to put the belt in tension, or to replace it.

Sounds like one of the auxillary belts slipping it may just need tightening up. I think the 4-Runner has two belts one large and one for the steering pump

Put a new fuel filter on and see if that helps, if it don't, better get a fuel pressure test.

Try removing the belt and turning each pulley by hand. If everything is smooth and free, with the belt still off, drive down the block once and see if the noise is there (this will eliminate all rotating accessories on the front). If you hear the noise, then there is a problem deeper inside. Oil pressure is good? Dont' forget when doing this, you will have no power steering, so be careful.

The noise is most likely the serpentine belt. Check it's condition and that of the belt tensioner. It sounds like the belt is slipping, causing the squealing noise u describe. As the belt warms up, it's getting a better 'grip' and the noise stops. Turning on the heat also turns on the a/c in newer cars, this added drag on the belt (one more component to turn, the a/c compressor) would explain why it happens when u turn on the heat. Is the belt original? If yes, it's time to change it, and check the belt tensioner to make sure it's doing it's job of keeping the belt snug on all the pulleys. Let me know how u make out.