Who is pinging?

[Blacklight]

(1) You don't always get DC in the water announcement and often when get them, they are not threat. So, I figure if you sprint and turn at such an announcement you may be giving away your position. So, I presume just getting this announcement is not a good time to do something.

It's a gamble but the secret is to sprint for only a little bit while the charges are going off and immediately throttle back. The sound of the DC's exploding should mask you for a little bit. Make absolute sure that you are as deep as you can get and have silent running enabled.

(2) Sometimes if you are listening you can hear the DCs hit the water and the volume is a good indication of the distance. Do you react if you get a load splash?

I sit and sweat until the boom like the sub mariners did.

(3) In other games and it seems hear too (but not so much), you can hear faint screw sounds from above. Should begin your DC evasion if you hear such noises?

Yes. If you can position yourself any direction away from that sound as best you can, you will be in a better position to dodge those big cans.

Here are some tricks that I do:

If you're deep and there's a destroyer coming straight in on you (You can tell on the hydrophone), turn directly TOWARD him. As soon as you hear the splashing of the DC's above you, go flank. You'll be behind him at that point where he can't hear you or ping you. Use this to your advantage and try to position yourself away from him. Going SLOW and using silent running is the key.

Another trick is, if your heckler is with a convoy or group of ships, try to get directly under the convoy or put the ships he's escorting in between you and the destroyers. The sounds of their engines will hide you.

Stay deep, stay slow, stay silent. That's the way to do it.
I'm sure others here will have more techniques to add.