In the pharse "в последнее время во многих почтовых отделениях начали открывать."
в последнее -> "Lately" appears to be set in accusative, direct object referring to time and if I remember correctly is often appears efter "B" like the locative.
and
"во многих почтовых отделениях"
It appears to me that it is pluralis with locative but the difficult part..is the "во" that sets the words in locative pluralis?
"Spatial" в takes the prepositional while "temporal" в takes the accusative.
во has nothing to do with it; the reason for it (and for those extra Оs on prepositions in general) is that the late-medieval sound shift that eliminated ultra-short vowels ignored word boundaries between a preposition and the following word. Thus medieval Russian въ мъногыхъ was affected, as if it were a single word, by the law that required that when you had two ultra-short syllables in a row, only the second vowel disappeared, and the first one became full-length (same as why you have сон < сънъ but сна < съна). It occurs highly inconsistently in modern Russian, but tends to survive with very common words such as многие or все.
"в последнее время во многих почтовых отделениях начали открывать."
Yes, ""в последнее время" is assusative. You can encounter it without a preposition, too, with the same meaning.
"во многих почтовых отделениях"
The preposition "во" is the same as preposition "в", and acquires the extra vowel when used before words starting with multiple consonants or with letter "в":
во времена...
во многих...
во праве (although 'в праве' can also be used)
Sometimes in poetry it replaced the preposition 'в' because the rhythm calls for it.
Во глубине сибирских руд / храните гордое терпенье ...
And, no, the preposition does not dictate the plural vs singular. And, yes, in this instance it's prepositional case. "Почтовые отделения" do not have a form of locative different from prepositional.
В последнее время is in Accusative. "В" is used with either Accusative or Prepositional case. Here you have Accusative.
Во is just the same as В. It's used to avoid spelling too many consecutive consonants, as it would have been otherwise: "вмн огих". No more magic here.
