to be honest, i don't understand why "quid" is the direct object in the sentence...maybe it's explained later in the book, but til now it does not say anything about it, except for nouns / adjectives in cases
The book does cover this in enough detail for this point in the language.
First, look at the section on the Dative Relation, § 43.
Here you are told that the relation is expressed by the preposition "to/towards/for". However...you are immediately told that it does not cover motion towards. You are given examples of such motion: "She went to town; he ran towards town; Columbus sailed for America." So keep in mind the idea that motion is not the dative case.
Now he describes the Indirect Object in § 44. The two examples given are:
1. The sailor announces the flight.
2. The sailor announces the flight to the farmers.
In the first sentence you can identify the subject object and verb. In the second sentence, they haven't changed but we've added an indirect object, "to the farmers".
Let's play a little substitution game here. The accusative case is in § 37 and tells us that it answers the question "What?" so everyplace we ws have the accusative case, we'll substitute the word WHAT.
1. The sailor announces WHAT.
2. The sailor announces WHAT to the farmers.
You can see how to turn the first sentence into a question:
1a. WHAT does the sailor announce?
If you use the table from § 33 you might think that the farmers answers the question WHOM...except for that preposition to. TO WHOM is what § 44 tells you the dative case covers.
2a. The sailor announces WHAT TO WHOM.
Now let's take the current sentence: quid ancilla equo legati dat?
First let's use our substitution, which in this case is really easy because we are given the WHAT.
WHAT does the maid give?
From our example above, we can turn this question back into a statement:
The maid gives WHAT.
So quid is the direct object.
Now the rest...
The maid gives WHAT TO WHOM.
TO WHOM does the maid give the WHAT? equo legati. indirect object, dative case.
OK, now the real problem sentence: cuius equum ancilla curat ?
Ingrid has mentioned to you that curat takes the accusative. This is mentioned in the text as well but you may have missed it. At the beginning of § 77 you are told to learn the special vocabulary. In that vocabulary you are given the verb curat and told its meaning is to care for and you are told that it takes the accusative case. That last part is the important part.
When you are given a word definition and told that it takes a particular case you are being directed to use that case over another according to the rules of the language. In this case, the verb carat, to care for, has a preposition built into it. so there is no preposition "for" to add when it is used in a sentence, like the one we have.
The made cares for WHOM?
Back to § 37 and we find the accusative case answers the question WHOM. So the definition of the verb and the sentence match up.
This leaves us with cuius, which you were given in the first special vocabulary as whose. So you end up with:
WHOSE WHAT does the maid care for?
I hope this helps.