(1) With any kind of transmitter, to get consistent output, you need consistent input levels, so that's the first thing to check. If the level in the second file are significantly different, it would be worthwhile to adjust it. This transmitter doesn't appear to have any way to accommodate variations in input level, so the easiest thing to do is try adjusting the output level from the PC playing the file. Commercial broadcasters use special hardware to automatically adjust levels into the transmitter, a technique called compression in the audio world. If you play a signal with a lot of dynamic range into this transmitter, you may find that the level setting that makes one part of the song sound better will make louder/quieter sections sound worse.
(2) That's definitely not amplitude modulation. It's essentially pulse width modulation. If you LPF pulse width modulation, you can recover an approximation of the input signal. An AM receiver will respond to the fundamental component, so apparently the fundamental's envelope closely approximates the input, but I'd be surprised if there wasn't significant distortion. You wouldn't call it high fidelity, but you could probably recognize voice through it. Using the 555 to do pulse width modulation is going to be especially sensitive to input levels.
(3) Transistor Q1 is probably not doing anything helpful. As shown, Q1 will be either completely off or .. completely off. The only signal getting to the antenna will be from capacitive coupling between the base and emitter. Two alternatives to try - you could just connect your antenna directly to pin 3, i.e., omit Q1 altogether, and get more signal onto the antenna that way, OR - leave Q1 where it is, but connect a 75 ohm resistor from its emitter to ground. This would make Q1 into an emitter follower (common collector) amplifier, and might give you some power gain, but pin 3 of the 555 is already a push-pull output, so just wiring pin 3 to the antenna would probably work just as well.