Newbie Arduino DC motor problem
I see a couple of issues here, which are related to your questions. I will use numbers here, but these do not necessarily correspond to the numbers you have used when asking.


The first thing you need to do when controlling a motor is to know it maximum stall current. The stall current is the current that will circulate through the motor (and through the transistor) when you apply an external force to stop the motor's shaft. But it's also the same current needed for the motor to start when it's completely stopped. So, for dimentioning a transistor, you typically will select one that has, as a rule of thumb, a collector current two times bigger than the motor's stall current.
The 2N2222 has a maximum Ic of 800 mA (best case, since there are 600 mA versions as far as I know). Although you are not specifying much about your particular motor, a typical 3V DC motor may easily be asking for 500+ mA. Some small motors (specially bad quality ones) may have stall currents bigger than 1 A. 
If your motor is asking for, let's say 300 mA when continuously running, then, you are near enough of the maximum current of the 2N2222.
But there is something more important here: dropout voltage in the transistor. The 2N2222 is a bipolar one, so it will have a Vce superior to 1V (probably around 2V), which means that you are dissipating a non trivial amount of power in the transistor. That power is not useful for your motor, nor for your transistor. Here, if the motor has a 3V nominal voltage, the 2V dropout in the transistor is even desirable, but I don't see your calculations about the real dropout in your description. So I assume that you are just counting that there will be an approximate 2V dropout. That means that, if your motor is taking 0.5A to work, you are dissipating 2V x 0.5A = 1W (Watt) in the transistor. But the poor 2N2222 is a 0.5W transistor! But even if your motor is using much less current, let's say 0.25 A, you are sill using it at its limits. That will explain why it's getting hot.
So, my advice is to go for a bigger transistor, if you want to count on its dropout to do not apply a voltage bigger than 3V to the motor (which can really reduce it lifespan a lot), AND do some calculations to be sure that at the currents you are using, you will have the Vce that you want.
Another option, probably better, is to replace that bipolar old transistor by a MOSFET, but you will need to provide the apropiate voltage for supplying the motor, since if not you will get nearly all the 5V to the motor (due to the very low Ron on Mosfets, and thus very low dropout voltage), and that will be very bad.
Also, please note that the 500mA of the USB port are probably not enough for your setup. But even if you are powering your Arduino from an external power supply through its power jack, you will find a good idea to use common grounds but different Vcc sources for logic and for the motor.
Finally a short advice: be sure that the diode protecting the back EMF (the diode connected with its cathode to Vcc in your circuit) is a fast diode. A slow, 1Nxxxx diode is usually to slow for effectively protecting your transistor. If you are using for example an 1N4007, you can switch to a UF4007 (Ultra Fast).
Regarding your 560R vs 2K2 resistor, this exceeds the answer a bit, so my advice is to read something about base resistors in transistors, like this. Also something about open collector configurations may help.
Finally, the practice of connecting the base resistor to Vcc in this kind of circuit to test the transistor should be fine. But DO NOT CONNECT the transistor's base directly, use always the resistor.