Electronic – Is another diode needed when using a tip120 transistor

There are several problems with your circuit:

1. 2N2222A is inappropriate for this application. You are trying to control power, although you didn't say what current this fan takes at 7 V. A hovercraft fan sounds like it would take some power, so let's say 1 A to pick something (since you didn't specify anything). That's more than the little 2N2222A can handle.

2. Your circuit doesn't have enough gain. You didn't say what the digital logic level of this arduino pin 9 is, but let's say 5 V when high. Figure 700 mV for the B-E drop, which leaves 4.3 V accross the 1 kΩ resistor. That will only provide 4.3 mA base drive. To support 1 A collector current, the transistor would have to have a gain of 233. Maybe some parts in the bin will do that, but it's well beyond the guarantee.

   This problem is even worse at 3.3 V logic or with a proper power transistor. You mentioned TIP120, but then are using 2N2222A for some reason. That makes no sense.

3. The flyback diode is in the wrong place. The problem is that the fan acts like a inductive load. You can't shut off current thru a inductor instantaneously. When you try, like when your transistor switches off, the inductor will make whatever voltage is required to keep the current flowing, at least in the short term. In your case that means raising the collector voltage to the point the transistor conducts anyway, which is bad for the transistor.

   Put the diode in reverse accross the fan. In normal operation it doesn't conduct. When the fan is abruptly turned off, it gives the inductor current a safe place to go until it dies down naturally.

4. Use a Schottky diode instead of the ordinary silicon diode you show. Schottky diodes have very fast reverse recovery time, essentially intantaneous for your application. That means the switch won't get beat up and you won't waste current when it is first turned on but the flyback current is still circulating. This can easily be the case when a motor is driven with PWM.

To fix this, replace the 2N2222A with a IRLML2502 FET and connect the PWM output directly to its gate. There is no need for a resistor between he PWM output and the gate. These are great little FETs as long as the voltage doesn't exceed 20 V, which it doesn't in your case. Don't forget to fix the diode before you change the transistor, else you'll be changing it again.

1. A transistor is a different type of (active) component entirely, to a passive component such as a resistor/capacitor/inductor. They have many different characteristics listed in the datasheet, one of the most important being current gain. If you don't understand much about them I suggest you get hold of a decent introductory electronics book and read up about them. There are thousands of part numbers, but in many cases you could use many different options. In a circuit such as this most general purpose NPN transistors will do (2N2222, 2N3904, BC337. etc)

2. The zener appears to be used in order to ensure the voltage does not rise above 3.3V (the zener will start conducting above this voltage) thereby protecting the 3V phone input from overvoltage. You don't need to calculate this value, you simply need a 3.3V rated zener, of which there will be many part numbers to choose from (assuming this is what the circuit requires)

EDIT - regarding more detail on why the transistors can be swapped, it's hard to explain briefly. They do have similar specifications, and are available in the common and easy to use through hole TO-92 package. The circuit shown is not ideal, but for a rough shifting circuit it will probably work with some distortion (how well depends somewhat on the input signals DC bias level)
Since important parameters like current gain can vary widely with temperature and between parts of the same number, a good transistor circuit will not rely on these parameters but control them with some basic feedback (e.g. by using an emitter resistor in a common emitter circuit)
Any good electronics book will have a section on how to use a transistor, understand it's parameters and design a circuit that meets your specifications with them. The website All About Circuits is pretty good (see section 3 for semiconductors), and book wise this question and it's answers should be of use: Basic Electronics Book. With a few hours of reading and a couple of breadboard experiments you should begin to see things more clearly.