Where can I find projects that are more tutorial than "copy-paste"? Some thing that talks about the math and analysis of it as well.
For future reference, stack-exchange is not a general opinion forum asking for recommendation or what basic Google searches could entail. 

What your question is basically asking "How do I become an electrical engineer?" to design circuits beyond logic gates requires extensive training in:


Signal Processing
Circuit Theory
Microprocessors
Digital Logic


The circuit course you are working on provides a fundamental understanding of analyzing circuits. there are a few things you should pay attention to in that class:


How to write equivalent capacitor, resistor, inductor loads
Voltage/current divider
OpAmps


Using knowledge of OpAmps you can easily design and implement audio amplifiers. 

If you want to design audio filters you need a good grasp of Signal Processing/Fourier Transforms to interpret and analyze filters. 

LED light show. 

You are asking for quite a bit.

You could use op-amp/comparator circuits. Digital Logic (JK-flipflops/Latches to store memory) 

Calculator:

Use an ALU or micro-controller. Hell, you could use a microprocessor to implement almost all of these projects. 

Updated to OPs new question, notice the edit at the top of the page 

When designing circuits Engineers think of everything as Blocks. 

Each block has a specific function that then feeds to the next block to achieve an overall goal. 

a microphone amplifier broken up into three stages. 

1) decoupling - removes the DC bias (if there was any) from the microphone
2) Biasing - biases the transistor so that we can use the full range of the rails
3) Gain Stage - applies a gain to the signal

Understanding how each one of them works requires extensive study:

Decoupling - Signal Processing

Biasing - circuits

Gain Stage - Electronics/Analog Circuit Design (yes there are courses offered on that topic specifically)