so that (sic) here are my stupid questions.
The only stupid questions are the ones one hasn't asked or is afraid to ask, etc.
Q1. When someone is called "barren" due to advanced age, does it mean that she is known to be menopausal?
The pertinent verse is [b]ut they had no child, because Elizabeth was barren, and both were advanced in years. [cf. Lk 1:7 (RSVCE)].
Elizabeth was barren and advanced in age (scripture does not say whether she was menopausal or not but advanced in age may be indicative of it).
Perhaps both are mentioned to show how remarkable the miracle of John's conception was.
Q2. Does the above telling by the angel suggest due physical efforts from Zacharias? or does it suggest that the pregnancy shall be by the Holy Spirit or something else?
Pertinent verses are [a]nd when his time of service was ended, he went to his home. After these days his wife Elizabeth conceived [cf.Lk 1:23-24 (RSVCE)], therefore it seems after Zechari′ah and Elizabeth had marital relations with God intervening because Elizabeth was barren, and both were advanced in age.
The reason I am asking the above ones is to figure out what part of the story is considered a miracle. After the Annunciation, I imagine Mary talking about the Annunciation with Elizabeth enthusiastically because Elizabeth will understand Mary thanks to her own "impossible" experience. But unfortunately, I am not sure about the impossible part of the above story. Just can't imagine what Elizabeth experienced through.
The difference between Mary and Zechari′ah is that Mary believed the angel's message and only inquired how it would be accomplished since she was a virgin and from her words, since her marriage with Joseph was virginal (she knew how babies are ordinarily made). Hence Elizabeth's praise, '[a]nd blessed is she who believed that there would be a fulfilment of what was spoken to her from the Lord.' [cf. Lk 1:45 (RSVCE)].
By constrast, Zechari′ah doubted the angel's words because even though the angel foretold the birth of John the Baptist, he couldn't see how this was possible since he reasoned that [he was] an old man, and [his] my wife was advanced in years. [cf. Lk 1:18 (RSVCE)]. He wanted proof: 'How shall I know this?'
Please see also: Did Mary have other children?.