I am not sure where to put these issues besides here, since I am a new user. But I have looked for similar problems and I cannot find any.
Problems:
1. When accelerating slowing, meaning my foot presses the pedal maybe half an inch to an inch, there is a squealing noise coming from the front of the car under the hood. I tried to record the sound but I couldn't pick it up. I can only describe the sound as being similar to when your brake pads are low and you hear the rotating metal on metal squealing sound. And it matters not what gear I am in(its 05 AT btw), it does not make the sound in neutral or park though. Also this is a recent sound within the month starting a little bit after I had my tires changed. Pulleys seemed to be fine.

2. When I am accelerating quickly once I reach a high enough rpm the engine area makes this sound that sounds like there is an air bubble or air is releasing from somewhere. I am going to try and record it but I do not know how well that will work either. But that sound has persisted since I got the car last Nov.

3. This noise started maybe last Sunday from what I can tell, and I think I know where the noise is coming from though I do not know what the particular part is. That noise was easily recordable and I was able to take a picture of it (attached). And sound below.
rx8 sound 1 - YouTube

Any help on any of these things would be so very much appreciated. The mazda dealership charges almost $100 to look at anything on the car even if it only requires them to lift the hood, so thank you in advance.

 

The sound from youtube sounds like your airpump is done. The airpump operates upon startup when the engine is cold. You can either ignore the sound or delete the air pump (look for threads about air pump).

As far as the squealing, make sure that the lug nuts are tied within the correct torque specs.

 

The whining sound is the air pump. It usually operates for 20-30 seconds at cold startup, and you usually won't hear it at any other time because you will be moving and the road noise, tires and engine/exhaust overwhelm it. I would not worry much about it, as all the air pumps make noise. Yours may be a little louder than usual, but I doubt that a used replacement would make much difference to you.

 

Thank you, I will look into turning it off or replacing it.

This is the sound from problem number 2, near the end.
rx8 sound 2 - YouTube

 

This sounds like MIAC (marbles in a can). Take a look in MIAC threads.

 

Ditto on the MIAC thought...