The minimum value that I am avaible to put on the stroke's size property in the Layer Styles is ONE. Is it any way to give value ZERO to this property? I am going to put keyframes on the stroke's size.
I also tried to change the stroke's Opacity property to ZERO to fix the issue, but because I am also using dropshow in the same layer, it created blind stroke between object and shadow of object.
No. Not with AE's paint engine at least. You need to think of something else like applying a Stroke effect to a mask path. That ansd of course you may want to read up on using pre-compositions and all that. Your drop shadow issue is a clear indication of trying to do everything at once and getting unpredictable results due to not managing render order.
You cannot change or rearrange the rendering order in Layer Styles. If you want the stroke to render before the drop shadow and you must use Layer Styles just split the layer one frame after your stroke gets to 1 pixel then turn off or delete Stroke from the top copy. Problem solved.