he can trim anything that hangs over however he wants, but I don't think it is okay for him to just tear apart my trees, cut them down, or throw the garbage into my yard.
You are right. His damaging the plants on your side of the property line is vandalism, and his throwing debris from his property onto yours is likely to constitute illegal dumping or littering. His actions, to the extent that they make your property look bad from any public areas, may also put you at risk of being cited by the city for violating Tempe's Nuisances and Property Enhancement Code (there's a link on the Tempe Code Compliance website).
Good luck getting the police to enforce any criminal laws in connection with this; they are typically very reluctant to get involved in this kind of petty neighbor squabbling. You could try an exploratory and preferably anonymous phone call to the nonemergency number, just to ask questions about the things your neighbor is doing, but (a) they probably won't want to do anything and (b) you may not want them to do anything, since it might make your neighbor REALLY mad and start a war that spirals out of control.
If you sue your neighbor, not only will you make him more angry (same as calling the cops on him), you are likely to spend more on lawyer fees and court costs than you will collect in damages. The law tends to be reluctant to try to value things that are difficult to place a dollar value on, including intangibles like aesthetics, privacy, or peace of mind - and including the difference between the value of a large tree and the price of a nursery-grown replacement. So you might be likely to win only the cost of buying and planting replacement trees for the ones he kills or damages irreparably. But I might be wrong (I'm not a lawyer, much less one who practices in AZ), so maybe it would be worth a couple hundred dollars to get a consultation from a local lawyer.
Did this guy ever try to talk to you about your trees (amicably), or has he been a jerk from Day One?
Building a privacy fence or a wall on that side of your property might be a good option if you can afford it. I took a quick look at the Tempe Zoning and Development Code (also linked on that webpage), and if I understood it right, you'd be allowed to build a wall as tall as 8 feet, except in the front setback area. If you do consider this idea, definitely talk to the zoning code staff about what you would need to do to be allowed to build right on the property line instead of inside the side setback. Ideally, you and your neighbor could agree to jointly install a property-line wall and each maintain your own side.