1. Was Dumbledore not aware of the map at all before Barty Jr. told him (during all the time Fread and George had it)?
I think Dumbledore did not know about the map based on following data.
“My master sent me word of my father’s escape. He told me to stop him at all costs. So I waited and watched. I used the map I had taken from Harry Potter. The map that had almost ruined everything.”
“Map?” said Dumbledore quickly. “What map is this?”
Harry Potter and the Goblet of Fire, Chapter 35, Veritaserum
If Dumbledore had know about the map he would not have asked “What map is this?”. Secondly lots of wrong-doing of James Potter were not known to Dumbledore because of the Invisibility cloak. Sirius, James and Peter went out of school using the Invisibility cloak
“They couldn’t keep me company as humans, so they kept me company as animals,” said Lupin. “A werewolf is only a danger to people. They sneaked out of the castle every month under James’s Invisibility Cloak. They transformed... Peter, as the smallest, could slip beneath the Willow’s attacking branches and touch the knot that freezes it.
Harry Potter and the Prisoner of Azkaban, Chapter 18, Moony, Wormtail, Padfoot, and Prongs
At this point Dumbledore did not know about Invisibility cloak, which becomes clear based on
You. You have guessed, I know, why the Cloak was in my possession on the
night your parents died. James had showed it to me just a few days previously. It explained so much of his undetected wrong-doing at school!
Harry Potter and the Deathly Hallows, Chapter 35, King’s Cross
2. Did Dumbledore know about the creators of the map? (Sirius/Lupin could have told him, but only if he had known to ask)
Dumbledore did not know about the creators of the map!
I doubt whether any Hogwarts students ever found out more about the Hogwarts grounds and Hogsmeade than we did... And that’s how we came to write the Marauder’s Map, and sign it with our nicknames. Sirius is Padfoot. Peter is Wormtail. James was Prongs.”
Harry Potter and the Prisoner of Azkaban, Chapter 18, Moony, Wormtail, Padfoot, and Prongs
The titles on the map refer to the nicknames of these men in their Animagus forms, and, in Lupin's case, werewolf form. Dumbledore admits to Harry that he did not knew that Sirius was Animagus
“Last night Sirius told me all about how they became Animagi,” said Dumbledore, smiling. “An extraordinary achievement — not least, keeping it quiet from me.
Harry Potter and the Prisoner of Azkaban, Chapter 22, Owl Post Again
I don't think Lupin would have told Dumbledore even if Dumbledore had asked him, because he felt he had betrayed Dumbledore's trust.
Lupin’s face had hardened, and there was self-disgust in his voice. “All this year, I have been battling with myself, wondering whether I should tell Dumbledore that Sirius was an Animagus. But I didn’t do it. Why? Because I was too cowardly. It would have meant admitting that I’d betrayed his trust while I was at school, admitting that I’d led others along with me... and Dumbledore’s trust has meant everything to me. He let me into Hogwarts as a boy, and he gave me a job when I have been shunned all my adult life, unable to find paid work because of what I am. And so I convinced myself that Sirius was getting into the school using dark arts he learned from Voldemort, that being an Animagus had nothing to do with it... so, in a way, Snape’s been right about me all along.”
Harry Potter and the Prisoner of Azkaban, Chapter 18, Moony, Wormtail, Padfoot, and Prongs
3. Did he approve of Harry using it? (Maybe he did, just like the invisibility cloak. Maybe he didn't as Harry could get in trouble, especially when Voldemort was trying to lure him out.)
I think he did. Otherwise he would have had the map destroyed after the events of Harry Potter and the Goblet of Fire. He could have used some spell (like Age Line) to block students from getting into Moody's(Barty Crouch Jr.) office. Harry was allowed an extraordinary amount of license/freedom by the headmaster Dumbledore.
“D’you think he meant you to do it?” said Ron. “Sending you your father’s cloak and everything?”
“Well, ” Hermione exploded, “if he did — I mean to say that’s terrible — you could have been killed.”
“No, it isn’t,” said Harry thoughtfully. “He’s a funny man, Dumbledore. I think he sort of wanted to give me a chance. I think he knows more or less everything that goes on here, you know. I reckon he had a pretty good idea we were going to try, and instead of stopping us, he just taught us enough to help. I don’t think it was an accident he let me find out how the mirror worked. It’s almost like he thought I had the right to face Voldemort if I could...”
Harry Potter and the Sorcerer's Stone, Chapter 17, The Man With Two Faces
