Assistance needed on what machine learning approach to use
Just a few observations from my side:


I think you should clarify your goal for yourself. Writing a program isn't a goal in itself. I'm missing the top level picture.
The second part: 'The goal is to have the program automatically tag similar sounds with the correct label by finding related sounds using machine learning.' is more like a goal, but where is the subjective part in it, and how does this answer questions in your thesis?
I'm not sure about the meaning of the output of Algorithm 1. Is this supervised or unsupervised classification? If it would be supervised, the 'supervision' is not subjective (until Algorithm 2). If is it unsupervised, the classes will find similarities, but the similarities are unlikely to correspond to the labels you intend to use.
However, if you take e.g. $k=9$ classes with k-means (you will have to optimize the number) the similarities in the classes will increase, and you could 'learn' the matching of labels and the classes. The $k$ classes should map to one of 3 labels.
Don't try to fit all these steps into one model. Create a few and connect them properly. Measure, interpret and report intermediate steps in your thesis.