Hi Lex,
I can only give some explanation here as it's beyond the scope of a forum to teach 101 electronics.
Start to search the net using keywords such as Diode clippers. or Audio design with opamps.
Anything you don't understand just search the net as there are mountains of info to be found.
some are better than others.
As your amp uses valves then some very good info can be found at the Valve Wizard site.
If you like the site you can purchase his book, it's very well written.

As for SS amp designs Teemu's book has tons of good reading that will help,, it's free and can be found right here on SSGuitar,,https://www.ssguitar.com/index.php?topic=711.0
regards to your questions;
Yes Audio path runs left to right.
In Clean mode SW1A and SW1B are open therefore VR1 is a 10k fixed resistor which sets clean gain of IC1A.
With SW1A closed (OD mode) VR1 becomes a Variable resistor (a Pot).
maximum gain is when VR1 grounds R5. (the CW (clockwise) sign tells us which way the pot turns.
Ok now the signal is much bigger so after the signal passes through the triode V1A it runs into the diodes D3,4,5,6.
SW1B has also been switched on as it's part of SW1A so now the diodes will clamp/clip/distort the signal at the threshold of the diodes set by the amount of diodes in the string. When the Gain pot is high the signal is clamped by the diodes. The amount of clipping is set by the number of diodes and the value of R14.
If you want less distortion make R14 larger. or As I mentioned replace R14 with a pot wired as a variable resistor then you can crank the Gain pot and then dial in the amount of clipping you want.
Hope it helps, Cheers,, Phil.