The words begegnen and kennenlernen are different verbs. While both have similar meanings, they go with different cases. It is
jemandem begegnen (Dativ)
jemanden kennenlernen (Akkusativ).
I am afraid that you will have to learn the case together with each verb.
Note that there is another grammatical category for verbs that is related, but not ...
The standard rule is always the same:
Dative case for places
Accusative case for directions
So, when you see a balloon hovering above a building, then you are looking at a place above the building, and then you need dative case:
Ich sehe den Ballon über dem Gebäude.
I see the balloon above the building.
But but if you are on a hill near the building, on ...
No. Verbs and prepositions govern the case. The verb sein demands the nominative:
Meine Mutter findet, dass ihre Ärztin gut ist
On the other hand etw. haben or etw./jmdn. mögen demands the accusative case.
In German, there are bestimmte Artikel (definite articles) and unbestimmte Artikel (indefinite articles). For example
Er zieht den Schuh an.
refers to a specific shoe and therefore uses the definite article "der" (in the accusative form "den").
Er zieht einen Schuh an.
refers to about any shoe, not a specific one, and therefore uses ...
No, it is not the rule that a noun ends on -s or -es being dative.
Those are genitive endings.
Dative nouns have either no ending in singular as your book says, or the archaic -e for masculine and neuter nouns.
In plural, there are several dative endings (e.g., -en, -ern).
A full translation is:
If you are finished, place the card on the table with its face side downwards and shift it to the person on your right (hand) side.
So the missing lowercase sie means the card (accusative object), Person refers to whoever sits on your right side (dative object), and zu Ihrer Rechten is a location of that Person given in its usual ...