I'm taking a stab at this; I'm not a native Spanish speaker.
I believe the aquí could go at the beginning, in the middle, or at the end. However, because the ellos is at the beginning here and the ayer is at the end, it would be cumbersome to put the aquí anywhere else but the middle in this sentence. That is, I think you could put the aquí after hijo, but it would just be awkward with the ayer right next to it. And if we didn't have the ellos (just the dejaron) it would sound OK to put the aquí at the beginning, but awkward with the ellos there.
I will try to get a native speaker to weigh in.
I could also take a shot. in Russian we also has this optional word order in sentences, and when you move words you just slightly change either meaning or accent in the sentence. in this case if I translate literally, in Russian sentence has accents on 'aquí' and 'ayer' which has to have crucial meaning in the text. just MAYBE it is the same way in Spanish
The indirect object pronoun is required whether or not there is a phrase to clarify it, but the direct object pronoun replaces the noun. In most cases, the direct object pronoun is "lo" for "him" for "it" and for "you" formal version, but in some places in Spain "le" is used for "him" for the direct object also, as well as for any masculine formal you. So, when there is both an indirect object and direct object, they use "se" for the indirect object to avoid alliteration. http://spanish.about.com/cs/grammar/a/objectpronouns_2.htm
The personal "a" in front of the direct object does not mean "to" which also looks like "a" but is in front of the indirect object. The personal "a" is like a marker that a person or a pet is coming next, but is not translated into English. http://spanish.about.com/cs/grammar/a/personal_a.htm
The verb let/allow in that sense requires another verb, so your sentences are incomplete, it would have to be something like "they let my son stay here yesterday" or "they allowed my son to eat here yesterday". Similarly in the Spanish, although the verb dejar can also mean let or allow, it only does so when coupled with an infinitive, for example "ellos dejaron comer aquí a mi hijo ayer" (I think). Good question though!