I suppose we are talking about Christian Europe during feudalism.

Yes, the year would be mentioned several times a year in church. There was an importance given to dates, for example the year 1000 apocalypse that didn't happen was an interesting chapter in history.

There were also other needs, for example founding dates and dates until which certain people do not have to pay taxes. Lhota, Wola, Wygoda, Sloboda (meaning time, benefit, freedom) are extremely popular village and town names in parts of Europe. CZ alone has around 300 of them. The name is based on time given by the feudal lord the the subjects in which they had no or low taxes and certain freedoms since the founding of the settlement.

Yes, we can safely assume that christians and jews had tradition of keeping knowledge of the current year and had even practical uses for it. So an average peasant would know it is Easter AD 1247, for example.