Where are you looking for these products? The Atmega8 can be found at a lot of different places online. Such as on digikey.com and search for Atmega8.
The servo is also easily found, but I don't feel like searching for it. However, in the $50 robot, using 2 DC motors would be A LOT more complicated than a beginner would probably want to deal with. It involves making at LEAST 2 more circuits that you will have to solder unless you buy them separately, however they are generally quite expensive (for a motor controller at least). (Yes yes I know you CAN find cheap ones)
Technically, NO, you do not NEED to solder, IF AND ONLY IF you buy a breadboard. A breadboard is a NON-PERMANENT circuit testing device, however it could be used to temporarily make a robot. Also, expect to spend more than $50 bucks if you want to go that route.
Learn to solder before you ruin any of the parts you buy, then you won't have to worry about not knowing how to solder again. There are correct ways to solder, and very incorrect ways as well. Google how to solder and learn away!