Alternatively/additionally, if I am commonly known as Ibn Yusuf after my father or even Ibn Ibrahim after my great-grandfather, is this a moniker I would have had from birth? How would those around me choose whom to call me after, and would this change once I’ve had my own son?
The answer to this is really simple, you would become known as "Abu Ahmed" as soon as people know that you have a son named Ahmed. In arabic naming conventions "Ibn" does not get used as a Kunya. Kunya only refers to Abu. The kunya is added to your name later in life either by yourself or a title someone else gives you. It does not necessarily need to refer to your first born male child.
A few examples:
The simplest one is that you take the Kunya based off your first child. So your first child is ahmad, you become Abu Ahmad.
If you have no children, or no male children, you can take the name of your father (or ancestor) as your Kunya. So for your example you can become Abu Yusuf or Abu Ibrahim. You would not use ibn (son of) because it does not qualify as a Kunya, in pre-modern arabic naming it would be equal to last name. For example: Your name is Ismail. Your son is Ahmad. Your Dad is Yusuf, grandfather is Umar and greatgrandfather is Ibrahim. Your name would be Abu Ahamd Ismail ibn Yusuf ibn Umar ibn Ibrahim (etc.) and you would be "Ismail ibn Yusuf ibn Umar ibn Ibrahim" from birth.
If you choose neither of the above you can use a hypothetical or metaphorical reference. Lets say you are a blacksmith. You can take on the kunya Abu Shakoosh. Shakoosh means hammer and since you are a blacksmith and you probably use a hammer you are now the father of hammer.