Why was Tom allowed to sit-in at the meeting. I thought he wasn't supposed to be hearing that type of stuff anymore, at least at that point in GFII.

Mike included/excluded Tom as an instrument of his manipulation of Tom.

Although it had to do with Roth, that meeting did not involve Michael's "new" business directly; it dealt with NY and the old family division of assets. If you notice, Mike excluded Tom from all illegitimate business meetings for his future plans, but kept him in meetings that included old business.

The exception of course is Tom handling the Whorehouse business with Senator Geary; that I suspect was because there was no one else that could deal tactfully with the shaken senator and yet not to make it too obvious that it was a set up.

This of course not withstanding Olivant's comment on manipulation.

Frankie was "old business"; Johnny Ola represented "new business." I think Michael wanted Tom in the Frankie meeting as a way of connecting Frankie to Vito, to whom Frankie had been loyal, and probably, as Olivant suggested, to placate Tom by keeping him in the action, so to speak.

Of course, Tom was also front & center in the initial meeting between Michael and Sen. Geary. (Which ironically, Geary thought would consist of only himself and Michael.)

I suppose that would also be considered 'new business', but not in as much need for privacy as the one w/ Johnny Ola from which Tom was excused.

Apple

I don't see the old business/new business dichotomy. Certainly, Neri and Lampone participated in old business. I see the dichotomy more between my people and Dad's people, Tom being one of Dad's people. However, in the end I think that what Mike permitted Tom to particpate in was a function of Tom's failure to protect the family whether expressed by Mike as a lack of confidence in Tom or as humiliation of Tom.

Tom would have been a natural to sit in on a meeting where Michael was just trying to keep Frankie in line becaue Frankie remembered Tom from the old days, and it would reinforce Frankie's sense of loyalty to Vito, and the Corleones in general. BUT this issue was not just some "street" "olive oil business thing." It had to do directly with the Rosatos, and Mike tells Frankie that he doesn't want trouble with the Rosatos because he is working on a deal with Hyman Roth. After this the whole scene turns into an argument about Roth, about giving his loyalty to a Jew over his own blood, etc.
Now at this time Tom knew that Johnny Ola and his men had come to Tahoe, and that Michael specifically tossed him out of the room so Mike and Rosato could talk privately. Tom may have had some weak points but he was no idiot. Here Michael is telling a loose cannon like Frankie (not to mention within earshot of Cicci) that he has big plans with Roth, and he doesn't want some street thing to get in the way. Tom would have easily deduced that whatever was going on with Roth had to relate to the Ola meeting earlier in the day. More telling is that Fredo was not in this meeting. Why not? Was he already being routinely excluded from all business meetings or did Michael not want him there when they were talking strategy regarding Roth.

Further, after the attempted hit Tom would have easily deduced it was either Pentangeli or Roth ...not Neri or Rocco.