Is a mini bread board a suitable replacement for a prototyping board?
Yes a mini bread board will work but if you want to get into robotics i would recommend to get the next size up from the mini as it is really usefull later on and mini might not cut it.
A prototyping board is better for the loong term as it can fit onto the bot however(upside down, on its side, shoved into the pare space) where as bread boards have to be kept in a suitable position to maintain good connections and to make sure the components dont get caught on something
but to start with a bread board will be fine. just take into consideration if it is worth getting a bigger one. its upto you.
Does it truely require no soldering?
Yep no soldering, the entire point of a bread board. very good for prototyping cicuits.
although you will need to solder your sensors.
Can i connect everything just like he did with the prototyping board and still get the same result?
Most likely not. A prototyping board has indavidual pads for each hole where as a bread board has rows of holes. The set out of which holes are connected etc depend on your board.(i'll post a pic)
I noticed he used wires on the back of his, is this just for organization or does it do something?
the wires connect the pins of the MCU to some components and headers etc. so yes it is for orginisation. you can put them on the top but when they are on the bottom it keeps the circuit board looking good.