Yes you can use the raw asparagus spear. I like to cut off the bottom just b/c the taste is a bit different. I only like to use one spear since it seems to overpower the overall taste if you put too much.
Aloe vera - make sure you get the food grade version if buying in a bottle (not the stuff you lather on after a sunburn!) Some health food stores carry the actual aloe leaf in which you can cut a small chunk (minus the skin) and toss it into your Bullet. Here's David Wolfe using it in his NutriBlast: