Quick question: At the end of Godfather II, Why does Michael start accusing Tom? Is it just Michael's paranoia getting him and Tom is the only one left to accuse? Besides the job offers, is there something that I am missing?

Accusing him of what???

I wouldn't say Michael isn't accusing him of anything but more like testing his loyalty. Michael knows he has to level the playing field and that Tom's feelings have been hurt along the way because he feels he should be closer to Mike on all the decisions. Knowing Tom could consider the VP position from Houstan's Hotels Mike asks him if he is going to come along with him in these things he has to do. Once Tom shows his loyalty all is well again.
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Several things are happening in that scene:
First, Michael had always acted as his own consigliere. He needed Tom after the Tahoe shooting. Now, he's putting Tom back in his place.
Second, he's acting on his dissatisfaction with Tom's performance. He left the family in Tom's hands before he departed for Miami, New York and Havana. When he came back, he found that Kay had had an abortion, and the Senate committee had nearly trapped him into perjury convictions because Tom failed to discover that Pentangeli was alive. You and I might not blame Tom for these things, but I think Michael's view was that "it happened on Tom's watch."
Third, Neri was making a move to displace Tom as Michael's number-two guy. Notice the smirk on Neri's face when Michael makes that speech about the job offer and Tom's mistress--it's clear that Neri was the source of that info.

Thanks! I knew there was something I was missing. Now that I know about Neri, it makes more sense.

I wonder if Neri knew that Rocco was never going to return from the shooting of Roth and his position would be cemented even further.

I always wondered what the significance was of Michael talking with Tom partly in Sicilian in that scene.

What was Michael implying?

He said, "So you're staying?" And Tom said, "Yes, I'm staying." I think Michael reverted to Sicilian to reinforce that this was about loyalty to the family.

I always found it odd that Neri had the balls to try and squeeze his way past Tom in the "rankings", Tom being a brother in everything but blood after all.

Would have been another potential storyline in between the end of Part II and Tom passing away, the Neri/Tom challenge. Unless of course, Tom had passed away by the time Michael was sitting all alone in the garden, dreaming of yesteryear......

Why would anyone find it odd? That was the nature of the business they were in. You paid your dues, scratched & clawed if you had to...to get a higher and higher position within the Family. Neri had been around for many years at this point and saw how Tom was being treated by Michael (though he may not have been made aware of all the reasoning behind it). He knew all along what would eventually happen to Fredo. The fact that Tom was Michael's brother wouldn't inhibit Neri in the slightest bit if he saw an opening that could move him further up and Tom further out.

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Or another "what if?": What if Neri whacked Tom in order to be Michael's undisputed #2 man?

It was to lay the groundwork for GF III. The MIchael-Tom Hagen conflict was going to be the major story line.

Related question: what did Michael mean in GFI when he said "you're out Tom". He wasn't out, but did Mike see Tom as a threat to take over the family and needed to tell him to back off?

I think it was just so that Tom was only involved in more legitimate things. Tom would never become Don or try and become Don

Two reasons, I believe:
First, as Hevy said, Michael wanted Tom to be completely "legitimate" so he could successfully front for Michael in Nevada, where Michael intended to be "legitimate."
Second, Michael told Tom, "You're not a wartime consigliere." I believe he meant that Tom lacked "Sicilian cunning" and "ruthlessness"--qualities that Michael may have felt could have prevented the Five Families War of 1946, Sonny's death and Michael's own succession to the Donship. In the novel, Tom blames himself for Sonny's death. "He was, he knew now, no fit consigliere for a family at war. He had been fooled, faked out, by theFive Families...Old Genco Abbandando would never have fallen for it, he would have smelled a rat...tripling his precautions."

I don't think so. The two films were more than 15 years apart, and, if I'm not mistaken, MP & FFC originally had no intention of making #3 until Paramount twisted their arms (and probably waved a lot of money at them).