Is there anyway to tell whether or not an adept is using an adept power? Are there any signs on either the physical or astral plane that this guy has Improved Skill(Automatics) while he's tossing lead your way? And does the use of adept powers leave an astral signature the way that spells do?

personally, I'd say other thatn the fact that he's hitting you more often, there's no way to tell. Same with whether or not you can detect kinesics, improved reflexes etc.
As to whether or not there is an astral signature, there was a significant thread a few months age so search for it. Again, personally, I'd say no, even powers that are activated, such as attribute boost. My reasoning is that 1 it doesn't state in the book that powers cause a signature, 2 there is no force associated with a power to determine how long they would last, and 3 adepts are physical, not astral.
Anyhoo that's my two worth.

I don't think so. Providing you have astral perception and the adept isn't masking, or you've managed to breach their masking, whilst you can certainly tell that they're an adept you wont be able to say whether they're using Improved Skill (Automatics) to help or just incredibly skilled mundanely. As for astral signatures I'd say no. The whole point of adepts is that the way they use their magic is internalised, unlike magicians who manipulate their surroundings and the local manasphere.

Edit: This is of course going by memory and without a copy of Street Magic to hand to check so YMMV of course.