Suspension on a cheap bike is, well, cheap. The Suntour, RST, or Rock Shox Dart fork you see on an entry level bike is going to bounce around like a pogo stick and sap up your power when you get out of the saddle and hammer. It's not until you get closer to $800 that you start to see decent forks that actually work the way they're supposed to on mountain bikes. In the mountain bike forum, the cheapest fork they recommend is the Rock Shox Tora, which will run you around $150 for the fork alone.
If you cut over a sharp enough object (broken glass?? What's a sharp object?) you'll put holes in your tires no matter what kind of bike it is. A cyclocross bike will be able to handle some rocks, as long as it's not technical trails or drops/jumps or anything like that, although I've seen cx bikes on some decent trails, albeit ridden by skilled riders.
Drops give you different hand positions (hoods, drops, tops of bars) and will be faster since you can get lower. IMO they're more comfortable for riding on the road anyway.
Relaxed geometry simply means you're in a more upright position. Compare the
Specialized Tarmac with the more relaxed
Specialized Roubaix. The easiest thing to see from the picture is the difference in saddle to bar drop, though there are other differences too.