Your circuit is incomplete and has some errors, but once those are fixed, it should work ok as an FSK transmitter once it is tuned up. For starters, let's talk about the input to the oscillator circuit. By this, I mean the audio ac voltage that is presented to the base of the 2N3904 transistor. The amount of frequency swing, or frequency modulation, that you get is called the FM Deviation. This Deviation is proportional to the input audio voltage. So, as you present an audio AC voltage to the base of the 2N3904, your carrier frequency will deviate up and down following the input voltage. The amount that it deviates up and down is important as this sets the bandwidth of your broadcast signal, which in turn is important because FM receivers are designed to hear signals with a specific bandwidth. That bandwidth is roughly 150KHz. The amount of deviation we get from your transmitter divided by your ac input audio voltage is called your modulation gain, with units of KHz/Volt. Unfortunately, we don't exactly know what the modulation gain is for this transmitter circuit you have selected, and the video does not reveal this, so we have to make an educated guess. I hope other readers who have built this circuit will chime in with their experience, but I am going to guess that the right amount of audio ac input is much less than 9 volts peak to peak in order to get the right deviation for a bandwidth of 150 KHz. I will guess that the amount of audio needed is somewhere around 200mVac peak to peak, but this is just a guess. In order to make it easier to tune this later on, you should add a voltage divider circuit to feed the audio to the base of the 2N3904.
So, you will have to add two resistors to form this voltage divider. One resistor will be in series with the transmitter input capacitor and the other resistor will follow this one and be connected to ground. The output from the junction of the two resistors then connects to the coupling capacitor feeding the 2N3904 base. This coupling capacitor blocks DC and insures that the bias to the 2N3904 is not altered.
Now, let's talk about errors. First of all, when I compare your transmitter circuit to the one in the video, I notice that yours is missing a capacitor from the 2N3904 emitter to ground. Put that in. Next, you show a connection from 9V to the Q output of the LM555. This will guarantee that the LM555 won't work, so you have to remove that connection.
Next, it is necessary to develop your circuit further by adding in names for each component (replace the question marks), and assign values to them.
Now some attempts to answer your questions further:
1) In order to modulate the transmitter, it is only necessary to superimpose an AC voltage onto the bias voltage present at the base of the 2N3904. Yes, in effect you are putting a voltage divider at the input and yes you can replace this with an oscillating voltage input and expect it to work.
2) as explained above, you don't want to turn the transistor completely on and completely off and in fact you want a much lower input level than a 9V swing, and this is why we must insert a voltage divider to reduce the amplitude from the LM555.
Now, this next comment is not meant as a criticism but rather a suggestion. It is possible to eliminate the LM555 circuit completely from this and still get it to work. They way you have it designed right now, the Arduino turns the 2N3906 on and off, which in turn causes the audio oscillator to change frequency from one audio frequency to another. This is generating AFSK (frequency shift keying) where a binary 1 is one audio frequency, and a binary 0 is another. The output of the Arduino is a sequence of 1's and 0's representing your data stream. It is possible to generate the audio frequency from the Arduino instead. You would have to program the D2 output to swing up and down at one of the two audio frequencies when a 0 bit is to be sent and have it generate the other audio frequency when a 1 is to be sent. Of course, what this means is that your program has to run fast enough to toggle the D2 output up and down at the two audio rates, but I think this is well within the capabilities of the Arduino. But you can save this idea as an enhancement step for later on, if you like.
You can include a couple of additional parts as insurance to protect the Arduino and LM555 from being disturbed by the high frequency AC. What you must do is include bypass capacitors from the 9V rail to ground at each IC. These capacitors must be connected as close as practical to the actual leg of the IC on the VCC side and on the ground side to be most effective. Use ceramic capacitors of something like 10 nF or 100 nF for this. The value isn't critical as long as the reactance of this capacitor is low (as in less than 2 ohms) at the carrier frequency.
Once you have made your changes, post your circuit again (with names assigned to each part so that we refer to them more easily) so we can comment.