What is the methodology behind 555 timer design?
The 555 was designed in the late 1960s - early 1970s, then the usual design method for analog ICs was in a nutshell:

think up and draw the block diagram

translate the blocks from the block diagram into discrete transistor based circuits (resistors and capacitors can be used as well). For some situations you might want to use special test-ICs that contain the real transistors as they are made in the actual IC production process (for high speed or low noise design, this can be crucial, not for an NE555 tough).

Build the circuits from 2) and measure them, do calculations, tweak it until the performance is good enough.

Draw the final design of the circuit as it is going to be on the chip.

Draw the layout and have that processed.

Measure your new IC and see if it does what it is supposed to do. If not, find out what is wrong, fix the design and try again.


Source: the columns from Bob Pease here and other stories I read some time ago on how gurus like Pease, Widlar, Gilbert etc. etc. worked long ago.
How do we do that today?
In a very similar way but the &quot;building your circuit with discrete transistors&quot; has been replaced by using a simulator and a PDK (Process Design Package) from the chip-foundry (factory where the ICs are made). So we design with &quot;virtual transistors&quot; as making an IC is expensive and takes a lot of time. In a simulator we can &quot;play&quot; until the cows come home ;-).