Learn German: Fragt er das Kind? - Is he asking the child?

In German, you form a yes/no question by putting the verb in the very first position (Verb-first or V1 order). Unlike English, German doesn't use helper verbs like "does" or "is" to make questions. You simply swap the subject and the verb: Er fragt (He asks) becomes Fragt er? (Does he ask? / Is he asking?).

The English translation says "Is he asking", but where are the words for "is" and "-ing" in German?

German doesn't have an "-ing" form (the present continuous tense). The single verb fragt means both "asks" and "is asking". You never translate the "is" or "are" when talking about an action happening right now!

Is das Kind the subject or the object here?

Das Kind is the direct object (in the accusative case) because it is the person receiving the action of being asked. The subject is er (he). Notice that the word das doesn't change! Neuter nouns like das Kind look exactly the same in the nominative (subject) and accusative (object) cases.

Why does the "d" at the end of Kind sound like a "t" when spoken?

In German, when a word ends in a "d", it is usually pronounced with a hard "t" sound. This happens with a few other letters too, like "b" sounding like "p", and "g" sounding like "k" at the end of words. So, Kind sounds like "kint".