Should I crush the spice for Chai Tea Latte?
The way we make Chai in India ( there are lots of variants of the spice mix) is to use the spices whole while brewing the tea and to strain the tea before drinking. 

Though for cardomom you should crush it to release the flavor. I would suggest using your hand or the broad side of your knife blade.

If you decide to use ginger, it might be a good idea to grate it to increase the surface area so that the flavor is absorbed quicker. (Ginger chai is really good in winter :))