How to achieve desired drawing in a new layer?

Here is an example with a layer mask, and a darker shadow color. You can get a more natural look by very slightly softening the edge of the layer mask. For blending mode, I used Linear Burn at 80%, but you will want to experiment to see what gives you the result you are looking for.

to make that layer look like a shadow, experiment with the blending modes. (start with Multiply, but one of the other modes may give a better-looking result.)

You're brushing in exactly the color you have chosen in the color picker, so I can't see anything wrong here. Naturally it's lighter, because you have picked a lighter color.

To protect areas you need to add a mask to that layer, so that it only affects the unmasked areas. 

Click the mask icon in the Layers panel to add the mask. Once the mask is there, a white mask reveals, a black mask hides. So target the mask by clicking the mask thumbnail, and start painting in the mask. You work on a mask just as you work in a normal image.

Target mask or image by clicking their respective thumbnails.