I'd like to address question #2. I do not think this is about the prophet from Deuteronomy 18.
1 Maccabees chapter 4:36-51
36 Then said Judas and his brothers, “Behold, our enemies are crushed; let us go up to cleanse the sanctuary and dedicate it.” 37 So all the army assembled and they went up to Mount Zion. 38 And they saw the sanctuary desolate, the altar profaned, and the gates burned. In the courts they saw bushes sprung up as in a thicket, or as on one of the mountains. They saw also the chambers of the priests in ruins. 39 Then they rent their clothes, and mourned with great lamentation, and sprinkled themselves with ashes. 40 They fell face down on the ground, and sounded the signal on the trumpets, and cried out to Heaven. 41 Then Judas detailed men to fight against those in the citadel until he had cleansed the sanctuary.
42 He chose blameless priests devoted to the law, 43 and they cleansed the sanctuary and removed the defiled stones to an unclean place. 44 They deliberated what to do about the altar of burnt offering, which had been profaned. 45 And they thought it best to tear it down, lest it bring reproach upon them, for the Gentiles had defiled it. So they tore down the altar, 46 and stored the stones in a convenient place on the temple hill until there should come a prophet to tell what to do with them. 47 Then they took unhewn[d] stones, as the law directs, and built a new altar like the former one. 48 They also rebuilt the sanctuary and the interior of the temple, and consecrated the courts. 49 They made new holy vessels, and brought the lampstand, the altar of incense, and the table into the temple. 50 Then they burned incense on the altar and lighted the lamps on the lampstand, and these gave light in the temple. 51 They placed the bread on the table and hung up the curtains. Thus they finished all the work they had undertaken.
This is the institution of chanukah and may be all, or part, of the reference in John 10 when at the temple for chanukah the Pharisees ask Yeshua why he keeps them in suspense.
Getting to the scribes and Pharisees asking John if he was the prophet comes in 1 Maccabees 14:41-49
41 And[m] the Jews and their priests decided that Simon should be their leader and high priest for ever, until a trustworthy prophet should arise, 42 and that he should be governor over them and that he should take charge of the sanctuary and appoint men over its tasks and over the country and the weapons and the strongholds, and that he should take charge of the sanctuary, 43 and that he should be obeyed by all, and that all contracts in the country should be written in his name, and that he should be clothed in purple and wear gold.
44 “And none of the people or priests shall be permitted to nullify any of these decisions or to oppose what he says, or to convene an assembly in the country without his permission, or to be clothed in purple or put on a gold buckle. 45 Whoever acts contrary to these decisions or nullifies any of them shall be liable to punishment.”
46 And all the people agreed to grant Simon the right to act in accord with these decisions. 47 So Simon accepted and agreed to be high priest, to be commander and ethnarch of the Jews and priests, and to be protector of them all.[n] 48 And they gave orders to inscribe this decree upon bronze tablets, to put them up in a conspicuous place in the precincts of the sanctuary, 49 and to deposit copies of them in the treasury, so that Simon and his sons might have them."
I'm assuming this is expecting the same prophet. Why this one is important to the Pharisees has to do with the fact that this is when Sadducees came into power. The prophet essentially boots them out of the picture.
To address question #3 in part. There was division in Judaism as to exactly what to expect from the messiah. There was division as to his role and whether because of this there were two messiahs in scripture. (This is evidenced in two places, in my opinion, through Caiphus and both sides used by John to indicate they were one and the same). There were also issues as to his nature as being purely human or divine in origin. This doesn't get mentioned, but the 4 gospels address these 4 issues. Matthew has emphasis on his Davidic kingship. Mark on him being the suffering servant. Luke's emphasis is on his humanity, and John hammers his divinity. Yeshua addresses this too in the response back to Peter, whose question you posted.
I don't have enough reputation to comment, only answer. I'd like to add this to the Elijah explanation you received.
From Matthew 16:11, 12
11 And He answered and said, “Elijah is coming and will restore all things; 12 I say to you, that Elijah already came, and they did not recognize him, but did to him whatever they wished. So also the Son of Man is going to suffer at their hands.”
John did not fulfill Elijah's role. If you pay attention to the narrative, immediately after Yeshua's baptism he began performing miracles that Elijah performed.
John told Jesus when he came to be baptized that it should be the other way around. Jesus said, "permit it at this time". To me, it appears Jesus stripped that role from him and took it upon himself. We then as believers take on the role of Elisha. In Matthew 17's comment by Jesus that now he is going to suffer also may be in connection with the curse of Malachi 4 which was a conditional prophecy based upon the response of Israel.