Rear derailleur's speed markings are generally for marketing purposes - if you're under 10 speed in Shimano, you can swap between road and mountain rear derailleurs without problems provided they have enough capacity for the rear cog combination you're using - the cable pull is the same, so road and mountain shifters can't tell the difference (often in fact, you'll see touring bikes or road bikes with big rear cogs with Shimano Deore derailleurs since the mountain derailleurs can have bigger cogs in the back at the cost of a few grams of extra weight and looking less cool). The speed number you can effectively ignore, since the amount the derailleur moves is determined by the amount the shifters pull the cable (in theory, the cage might be thinner or something, but in practice this doesn't really matter).
You also won't notice a difference in performance or anything if you just stuck a Claris rear derailleur on. It would just be a waste of money.
If you want 8 speeds, you need to buy an 8 speed cassette (Shimano compatible) and an 8 speed Shimano shifter for the rear. You can keep the existing derailleur - you just need to adjust it a bit. You'll need some special tools (cassette remover, cable cable cutter, big crescent wrench) in addition to hex keys in order to do this as well. Swap the shifter and put the cassette on, adjust the derailleur and you've got an 8 speed setup.
Personally, I think you should just stick with 7 speeds as it is sufficient for commuting (the extra speed doesn't really help - its just a bit easier to find 8 speed parts these days, so thats pretty much the only reason to go from 7 to 8). Spend your money on beer or something instead. The Cannondale Quick 6 is a relatively cheap hybrid (~500 dollars new), so if you want something faster, you're probably better off getting a different bike with probably more aggressive gearing and geometry (say a flat bar road bike like the Giant Escape or something if you want to stick with flat bars).