A brief education on USB 2 vs 3:
Think of USB as a pipe (It's actually called a "bus"....Universal Serial Bus) that can only push so much data through it per second. For USB 2, this limit is 480Mbps, or 480 Megabits per second, and for USB 3, it's 5Gbps...5 Gigabits per second. This means that across ALL USB 2 devices connected to your computer, the bus will at most support 480Mbps of throughput. Put another way, if you have 2 devices rated at 400Mbps connected at the same time, you will max out the USB 2 capability before you max out the device capability.
USB 3 is backwards compatible with USB 2. This means that your USB 3 drive WILL work when plugged into a USB 2 port. It will only operate at USB 2 speeds, because that is the bottleneck.
Now, to actually answer your question: As stated above, most USB 3 devices work in USB 2 ports. It will just be very slow. Your assumption that they both need to be USB 3 is incorrect.
As for whether you can run your machine from an external drive: yes, you can. You don't want to though. It will be VERY slow. To give you an idea, at work, we have some Seagate drives that are god-awful terribad slow, and those clock in at around 71MBps (capital "B" here; I'm talking in bytes not bits) read speed and 69MBps write speed. USB 2, however, only supports 60MBps (480 megabits / 8 bits per byte = 60 megabytes). Even these hard drives at work would slow down even more.
Now, think what it would be like if you wanted to use ANOTHER drive as a time machine backup. You just cut those speeds in half because the bus needs to be shared. Ouch!
As another point of interest, you expressed an interest in having your OS mobile. In general, this is not a fantastic idea, ESPECIALLY with a spinning drive. If you ABSOLUTELY need a mobile OS, I suggest you find yourself a 64GB flash drive on sale (Newegg had one for 18 bucks just the other day, with a lifetime warranty) and put your OS on that. Then, use a normal internal drive for all of your data. Then, back up your data to some OTHER external drive, or whatever, if you need to bring it with you.