How to calculate voltage of these nodes
You can use mesh equations for this (the most general and fundamental way), but the simpler way is to successively work your way from the bottom left up to the top right combining resistors in parallel and series.




The rule for series it that the same current must flow through both
components.
Similarly, the rule for parallel is that the voltage across both
components must be the same.


Therefore, R1 and R3 are NOT in series. There is some current flowing out through the middle into R5. You can't just ignore that. That means the current running in R1 is not the same as the current running in R3 which means they are not in series. The only resistors in series with each other in this entire schematic are R13 and R14.

That means, for example, R12 is NOT in parallel with R10, nor is it in parallel with R13 or R12. However, the series combination of R13+R14 is in parallel with R12.



You also cannot use the resistor divider equation here due to its limitations. That limitation is as follows:

The resistor divider equation assumes that the output node of the divider has zero current flowing out of it. That means that all the current stays in the divider. As soon as some current flows out the divider's central output node, the current flowing in the bottom resistor is no longer the same as the current flowing in the top resistor which then causes your voltage stepdown ratio to be dependent on the load.

In reality, this means you can only use the resistor divider equation to approximate the voltage at the divider's output when the load impedance is much much greater than the resistor values in the divider itself.