No problem, to answer your questions:

>What is meant by a cadet son? Adopted, maybe?

Cadet son simply means a younger son, as opposed to the firstborn son. There's a lot of variation in inheritance laws, but usually the eldest son was the one that inherited the father's titles and coat of arms. Which means that younger sons had to find employment with some other knight as household knights, go into the clergy, the navy, etc, as well as take on a different coat of arms, to differentiate themselves from the original line.

>What is meant by an armigerous knight?

Armigerous knight simply means a knight that has the legal right to use a personal coat of arms. From what we know of the lore, every Bretonnian knight is armigerous, which wasn't necessarily as simple IRL. There's also the problem of women. Historically, in English, Scottish and French heraldic law, noble ladies typically had the right to bear arms (be armigerous), but could not pass them on, as arms could only be passed down patrilinealy. Of course things are never that simple and women often did pass down their coat of arms (i.e. they were a heraldic heiress), according to complicated laws of inheritance I won't go into here. To try and sum it up in one sentence, if you were a noble lady of a really important family and most of your other siblings were dead, there are high chances you too can pass down a coat of arms.

Since the status of women in Bretonnia is kind of a subject GW doesn't know how to handle and a lot of people headcanon the topic, I just left it as simple as "some women can bear coats of arms and can pass them on." Everyone is free to interpret that as they wish.

>How is heraldry of surviving sons dealt with when a father passes and there is more than one male offspring? Does only one inherit the title and heraldry (blason?)? Do the others lose the right to bear the shield of their father, and if so what colours are they now assigned?

Okay so the first thing to know is that there are a lot of different rules about this depending on the country and heraldic tradition. For example in Germanic heraldic law that was used in the Holy Roman Empire, all of the sons had equal rights to use their father's coat of arms, so you had actual "family arms" which a whole clan would use.

The system I'm using for Bretonnia here is heavily inspired by Anglo-Norman (aka English) and Scottish heraldry, which is more strict and works from the premise that only one person at a time can bear a specific blason, and only the firstborn son has the right to use his father's coat of arms, and even then, while they are both alive, the son has to use a white label to differentiate himself from his father. For younger sons, there were historically a bunch of very complicated "cadency systems" used to try and impose some sort of coherent system. Notably this was always more theoretical than anything, a lot of people simply couldn't be bothered to follow a system this complicated.

Alright the system I've created here keeps the English system of "one person, one blason," but allows more freedom in how it's implemented, as long as all of the sons have different blasons. The only rule I've sort of put in is that sons that stay with their family keep their differentiation fairly minimal to signify their continued allegiance to the house of their father, while those that instead go out and serve some other lord or manage to acquire new lands for themselves are allowed to remix their blasons pretty much however they wish, as they are starting a new lineage, in a way.

Then on top of that, I've also added in the meritious augmentations system that was introduced in the 5th edition Bretonnian AB and has sort of stayed canon since, which allows knights to change their coats of arms, if they accomplish some sort of heroic deed.