Is there a way in Illustrator to know the exact dimension of shape in pixel and possibly to resize it down to different (but precise) dimension?
If you got multiple objects that you want to resize as one, you need to start by selecting all of those objects with the Selection tool.
Still with the Selection tool, in the upper right corner you will see the width and height boxes that you can use to resize your objects.
Alternatively you will find the same boxes from the top menu:
Window > Transform
If you want to keep aspect ratio, you need to click the chain between the boxes.