"개" is attached at the end of the word being counted. It is a count word and is used sort of used like pieces in the phrase "Two pieces of chocolate". But Korean just uses it all the time where in English it is only used occasionally.

아이 is child and you wrote boy. Most probably thats the reason

개 is a counter for the ball (공 한 개 object-number-counter) and 를 is the object particle since you are giving the ball (object)

Yes, that's how clauses with transitive verbs work in English; you can drop the preposition between the verb and its indirect object if both appear before the direct object in the sentence. "Give a ball to the kid" but "Give the kid a ball".

A native English speaker will intuitively understand the word order as meaning the verb is ditransitive, the first noun is the indirect object, and the second noun is the direct object. "Give him a donut", "pass the lady her change", "throw the dog a bone". You need to add "to" before the indirect object if it comes after the direct object to signal that the word order has changed.

에게/한테 is the dative marker; it is attached to the word at which the verb in the sentence is directed, i.e. whatever or whoever the verb is being done to. In this sentence, 아이 is being given 주다 something, so it is marked with -에게. 가/이 is the subject marker; this sentence is an imperative clause, a command, so the subject would be the listener (i.e., "You, give the child a ball," 'you' is the subject). In these kinds of sentences, the subject is often omitted, as it is implied in the structure of the sentence.

Forget it

개를 means not only a dog but also an unit. you are right. the sentences are very confused and they should give more explanations.