So I can only really speak for Medieval England and possibly Scotland. But given that narrow geographical focus you can be relatively assured that the majority of people would have known who their King was, and there is a major contributing factor. The Church at this time was really one of the only institutions capable of disseminating information to the masses of the rural population, as attending Church on sundays was a regular tradition and priests, monks and friars were relatively connected with the wider world. Thus most sermons would have included some information regarding the political situation of the realm when it was applicable, and the death of a King would be mourned throughout the kingdom primarily through church services. Information about a new King would be similarly dispersed.

Now whether they would care is a different matter, and one that might be a little hard to answer given we don't have a ton of peasant sources throughout the medieval period at all. Kings did have a significant influence on the Kingdom though - if you were a wool trader for instance, you would probably be peeved when Edward I laid claim to a portion of all wool revenue in the kingdom. If you were a peasant in the Welsh Marches chances were you would be drafted or semi-volunteered into military service, which could definitely help form your opinion of whomever King is calling for war.

It was certainly the Lords who would watch a royal succession the closest however, as you imply. Lords simply had much higher exposure to the royal family as well as understanding of the political/economic situation of the realm, and their prior relations with an heir apparent for instance could deeply affect their fortunes during his coming reign. Most would not allow a bastard to inherit the throne however, as you ask. While William the Conqueror was a bastard for sure, after his reign there was little to no chance of a bastard inheriting the throne of England. In fact it was often harder to stave off the increasing numbers of different legitimate claimants than to keep bastards off the throne.

Frank Barlow The Feudal Kingdom of England 1042-1216 Routledge; 5 edition 1999

Michael Prestwich Edward I University of California Press, 1988

In medieval Flanders and Brabant literally everyone would've known who their count and duke were. Even the king of France and the German emperor were generally known to people. These grand successions didn't really happen that much after all, and although news of it might travel slower in some regions than others, it most definitely did travel. In the bigger cities they most probably even had an idea of what their lord looked like and had probably even seen him before.

In these areas people most definitely cared about who their lord was, but mainly due to self interest. During problems of succession, the lord often made alliances with powerful guilds, who could mobilise a large amount of commoners in acts of rebellion. So although these commoners would often act in the name and interest of a lord, they did so solely for personal gain. That doesn't take away the fact however that for that reason they did care about who their lord was: preferably someone with decent policies towards guilds etc.

It is difficult to say to what extent average day labourers cared about these things, but seeing as they often also participated in urban revolts, sang songs about their lords, heard news about their wars and probably even felt it (physically, economically, ...) etc., they must have cared to a considerable extent.

They did however care less about their count or duke than they did about the urban government. The urban government was the embodiment of local rule in these areas, not the lord. So although they knew about their lord and often even allied to him specifically, they usually cared mainly due to the impact it could have on local government.

Thanks for the answers, really appreciated the time you took to write these!