"The man and the boy"
Translation:Der Mann und der Junge
das is used when an object is in the sentence
Do you mean "object" in the grammatical sense (direct object), or in the plain English sense of "inanimate object (not an animal or human)"?
Either way, it's wrong.
For example, the sentence der Mann sieht den Stein has a grammatical object (what does he see? -- the stone), and stones are objects, but we don't say das Mann sieht das Stein.
Das is for lifeless objects
That is incorrect.
Lifeless objects can be any grammatical gender in German -- for example, der Löffel (the spoon) is masculine, die Gabel (the fork) is feminine, and das Messer (the knife) is neuter.
And das can be for words referring to living beings as well, e.g. das Mädchen (the girl), das Opfer (the victim), das Pferd (the horse).
Grammatical gender is, in general, not predictable and simply has to be learned.
das Mädchen "the girl" and das Kind "the child" are grammatically neuter, der Junge "the boy" is grammatically masculine.
In general, you can't determine the grammatical gender of a word just by looking at it, so you have to learn the gender along with the word. (Similarly with the plural, which you generally can't reliably guess, so you have to learn it along with the word as well.)
It may help to memorise not simply Junge but der Junge.
The English word "boy" doesn't have grammatical gender.
The German word Junge is masculine.
(If you're asking about the gender of a German word, ask about a specific German word, please -- English words can sometimes translate to two or more German words, which can have different grammatical genders. For example, a table to put food on is der Tisch but a table of data is die Tabelle. Or the end of something can be der Schluss or das Ende.)
It's misleading saying that die is used für Mädchen.
It's true that die Mädchen is grammatically correct, but it means "the girls" (plural) -- plurals of all gender use die, e.g. die Männer (the men), die Frauen (the women), die Mädchen (the girls).
But in the singular, it's das Mädchen, because the noun Mädchen has neuter gender, not feminine gender.
Generally, yes, you repeat it every time -- especially since chances are that the two words will have different genders.
Also, der Mann und Vater (for example) sounds like you're talking about one person who is both a man and a father, so if you are speaking about two people (a man and a father), you would say der Mann und der Vater.
Case of the letter matters..?
In German: yes, letter case matters.
Duolingo ignores case when checking (unfortunately), but if you want to learn proper German, you should learn the proper capitalisation just as much as the proper letters that make up a word: both are part of the correct spelling.
sie and Sie are just as much different words as schon and schön, for example.