Exactly what kind of a deal did Michael want Frank Pentangeli to make with the Rosato brothers? There was a big shootout after that and I couldn't tell who was shooting who.

Michael wanted Frankie to act like he was willing to make peace with the Rosattos and work things out. Michael wanted this done so not to jeapordize his own deal that he was trying to make with Roth.

Don Cardi

This was one of Michael's cleverest ploys:
He showed up at Frankie's place yelling about the Tahoe attempt, making Frankie think he blamed him. Then, when Michael said he knew Roth was the culprit and asked Frankie to help him, Frankie practically passed out with relief. "Michael, anything..." he said. That's when Michael pounced: "Settle these problems with the Rosato brothers."
When Frankie went along with it, he made a giant concession to Michael: putting his own interests in the "three territories in the Bronx" behind him, and putting his own life in jeopardy by meeting with the Rosatos. Michael had strictly selfish reasons: He wanted Roth to think that their mutual deal was still on. If Frankie succeeded in settling his problems with the Rosatos at that meeting, so much the better for Michael--he'd have one less problem in NY. If the Rosatos killed or tried to kill Frankie, it'd be further proof that Roth was behind the Tahoe attack. Michael couldn't lose: clever Michael! Poor Frankie!

When Michael was suspecting even his close family members(except Tom), why the hell did he tell Frankie about all his plans(plus all the philosophy, his father's advice and all other miscellaneous wisdom )? What if pentangeli was involved indirectly?

because frankie wasnt close enough to approach for betreyal. It had to be someone close to Michael, who he would trust, someone who would be able to (more or less) tell Roth where would be the easiest place to hit Michael, but also someone who wasnt too loyal to Mike that they wouldnt say "Roth approached me".Five Angels wasnt close enough that he could be telling Roth were mike hung out at certain times of the day.

Also, Roth overplayed his hand:
Roth probably was invited to Anthony's party but bowed out because of "ill health." But he knew that Frankie would be there, contentious about the Rosato brothers and anxious for a private audience with Michael. Therefore, Roth figured, Frankie would have an obvious beef with Michael, and would be the obvious fall-guy for the hit, successful or not.
What Roth didn't count on was that Michael--far smarter than he thought--saw that it was all too obvious, too pat, for Frankie to have made the attempt. Frankie wasn't the sharpest knife in the drawer, but he wasn't dumb enough to think he could whack Michael and get away with it. That's why Michael concluded almost immediately that Roth was the culprit.
I said "almost" because that scene with Frankie in his study was a kind of loyalty test: Frankie passed it when he agreed to meet with the Rosatos.

Good point Turnbull.

Don Aprile

What kind of relationship did michael share with Pentangeli?
In the dinner scene, we see pentangeli throwing the glass on the table with disrespect. In michael's room again we see Pentangeli saying that he need not be told how to run the family. In both these scenes, pentangeli is not shown as a subordinate who cannot raise his voice against michael. But in Pentangeli's house michael shouts at him and pentangeli is immedietely meek and scared. What was his role in michael's business? was he a caporegime?

My understanding is that he wasn't quite a capo, but not quite his own boss. He took over Clemenza's territories,, but at this point Clemenza was supposed to have formed his own family, seperate from the Corleones. Based on what Michael says in GFII, however, this isn't the case.

Pentangeli was still firmly entrenched in the Corleone's business affairs, and he was still an indirect part of the Corleone family; while the main part of the family was in Nevada, Clemenza/Pentangeli took care of the family's "assets" in New York. After all, when he says he can run "his" family the way he wants to, Michael reminds him that his family still bears the name Corleone, and that he'll run it like a Corleone (in other words, do what the Godfather tells him to do like a loyal soldier).

I liken the situation to a large corporation: You have the Chairman of the Board (i.e. Michael) who runs the main company, and then you would have subsidiary companies with their own President and staff who can pretty much do their own thing, but are still technically under the "parent" company. (i.e. Pentangeli)

Frankie was a caporegime, like Tessio and Clemenza before him. His job was to run the "olive oil business" in NYC. He was subordinate to Michael.
Frankie wasn't disrespectful to Michael, but since he was in NYC and Michael was in Tahoe, he wanted more independence. Probably he disagreed with his boss more vigorously than he would have if Michael was still in NYC, supervising him closely.
Two other points:
1. Frankie didn't throw the wine glass--he knocked it over because he was drunk.
2. The reason he was meek and cowering when Michael confronted him at his home was that he feared Michael would blame him for the Tahoe attempt. Michael did his best to foster that impression by shouting at Frankie. So, when Michael told Frankie that he knew it was Roth, Frankie practically passed out with relief. Then Michael pounced: "I want you to help me take my revenge." "Michael, anything..." Frankie replied, blissful with relief. "Settle these problems with the Rosatos," Michael said. Now Michael got Frankie to go along with something he was arguing about just days earlier in Tahoe. It was Michael at his most brilliant and manipulative.

I agree he was drunk, but I also thought it was an utter show of dispespect. Look at how Michael and even Moma "C" look at him afterwards. They can't even believe he did that.