Electronic – arduino – Why is the voltage across a Zener diode so dependent on current

It's just the basic operating principle of a diode. An ideal diode would allow current to flow in one direction but block current flow in the other direction. This is based on how it's made, with a p-type region, an n-type region, and a depletion zone in between. Like the bottom diode in this picture:

When you apply a some voltage, in your case 0.9V then the p-type holes and the n-type electrons move into the depletion region because they are repelled by their respective battery terminal. With enough voltage (0.9V in your case) the free electrons in the depletion region get moving and current begins to flow like this:

Now in the ideal case if you were to reverse that battery the opposite will happen and you'll get no current flowing:

In the real world though you can only apply so much reverse voltage or push before you hit the breakdown voltage and current begins to flow freely in the reverse direction. Zener diodes take advantage of this fact and are constructed to breakdown at lower voltages such as your 3.3V.

Sources:
You can read more about how zeners are made here Or see the article I got all the pictures from here

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.