When Tom is talking to the Senator after the prostitute is butchered, Neri comes from out the bathroom cleaning a shot glass. Tom shoots him a look that seems to say "Hey, get out of here...what are you doing?"

I may be reading too much into it, but what was that gesture supposed to mean?

I thought the implication was that Neri killed the prostitute, so maybe it was a "Why are you still here? People might be suspicious!" type of look? Obviously the Senator was drugged, maybe he was removing the evidence?

...and washing his hands.
Yes, Tom gave him the "get out of here" look because Tom was about to talk to the Senator, figuring he was going to be semi-lucid. The last thing Tom would want would be for the Senator to see the real assassin.

That scene was very unique in a way. Obviously it was brutal because of what happens to the poor girl. But it also shows kind of the small rivalry between tom and neri. Neri is slowly becoming mike's right hand man which is what tom was so i think that look had some other undertones as well

What do you think: Was it Tom's idea to kill the girl?

Doubt it. Only mike is that evil at that point. Tom is no saint but mike is the only person who orders the murders of women and his family

But Tom is just as culpable for going along with it. Tom was also actively involved in horse's head incident when Vito was still in charge so I wouldn't underestimate him.

Not underestimating him im just saying that mike was the one who ordered it and tom probably wen to neri to have it done. Never said Tom was innocent but im sure mike was the one who ordered it. I mean he has ordered women killed before

I haven't read the book. What other woman did he order killed? I thought this was the only one. In GFIII, I saw him losing his own daughter as a sort of Divine retribution.

For all we know it could have been Tom's idea and Mike sanctioned it. They're both just as bad. When Vito sent Tom to take care of Woltz, it could have been similar situation. I don't think everything was micro managed by the Dons but I'm sure Tom would have needed permission for certain things.

Tom may have given the order to Neri, but the murder most probably was preordained by Michael:

Before leaving Tahoe, Michael told Tom, "You're gonna take over, you're gonna be the Don." But I doubt that he gave Tom a blank check to order a murder on his own, especially one that involved someone as important as Senator Geary. Michael had already set that murder in motion:

Geary was a regular at Fredo’s brothel, with that hooker: As he said to Tom, “We done it be-FOE-wer.” Fredo would have told his brother: “Hey Mikey, guess who’s a regular at my joint? Your pal, Senator Geary. “Yeah, he’s into some kinky s**t with the same whore alla time.” Michael would have tucked that fact away like a gilt-edged investment. So, as soon as Geary left Michael’s office after the “oily hair/silk suit” speech, Michael would have turned to Hagen and Neri: “Now’s the time play that whorehouse card and get that scumbag to give me the license. Find out when he next visits Fredo’s joint and nail his ass." They would take that as a standing order to drug him and kill the hooker, if need be.
As for Tom/Neri rivalry: We see it in the penultimate boathouse scene. When Michael humiliates Tom about the offer of employment from the House and Hotels and tells him, "You can move yourself, your family and your mistress to Vegas," Neary is smirking--you can tell he was the source of that info.

The girl with tattalgia was killed by Rocco at the end of 1. that still counts because mike knew that he would be with a woman so he probably said to kill her to

The woman was not the target. Maybe Tattaglia overslept and they weren't expecting to find him with company or Rocco was sloppy. I don't think Mike would be involved in that level of detail, but on some level he bears responsibility. To say that he ordered it would be too much of a stretch.

well they knew he was gonna be with a woman so its not that much of a stretch

I would suspect that Fredo mentioned to Mike that he knew Geary from the brothel, and the planning went like this:

Meeting in Mike's office

Tom: This Senator....how are we going to deal with him?
Mike: Fredo says he frequents one of his "houses."
Tom: That's perfect for us. We can compromise him with this.
Mike: Compromising him isn't enough. Besides he'll be able to call in enough favors to keep this thing quiet.
Tom: What if we caught him and made it appear we were covering for him?
Mike: No, he's too slippery. The only solution is to silence him with something more. Al?
Neri: If he thought he'd killed one of the hookers....
Mike: Ok Al you get in there and drug him, and kill the girl. If Fredo is right he plays some kind of bondage game, so you make it look like he went too far.
Tom: He's got to think that you found out about what he did through Fredo.
Mike: I don't want Fredo directly involved. Al you do exactly what you plnned, and once the Senator is coming around I want Tom to go in there and convince him that we have the ability to keep this thing quiet. Of course you also have to convince him he's guilty.
Tom: That shouldn't be too difficult.
Mike: Good, but it has to happen after I leave for Havana. I don't want to be in Nevada when this happens.

(Problem is Tom and Mike forgot to tell Al to get th hell out of there, or at least keep himself hidden so the senator would not see him).

That was FFC's directoral license at work again: he needed to show us that Neri killed the girl, otherwise we might have thought Geary actually did it. Too bad Geary didn't see Neri.

We are almost going full circle to the next Geary issue, nmely whether his speech and departure from the Senate hearing Michael "attended" was a favor to Michael or a way to get out from under his thumb.

As I posted many times:

Geary, as a member of the subcommittee, had to know that they were holding Pentangeli. So, when he asked Cicci the "buffers" question, it was a deliberate setup to make Michael think he could get away with perjury by believing that Cicci was the highest-ranking witness against him--and Cicci said he never got a direct order from Michael.

Agreed.

I've argued in the past that Geary is merely sitting in with the committee, and not actually a member. I have checked with someone I know who does legislative work, and feel pretty sure that this is a least a possibility -- if the chairman lets you, you can sit and ask questions. Given that Geary calls the other committee "my own committee" and that he leaves what would surely be the hottest hearings of the year, it seems likely to me that he was just there by invitation.

I've tried to figure out how many senators are actually sitting at the tables, as an even number would imply that someone was just sitting in, but have never been able to break it down.

A typical Senate standing committee would have roguhly 20 members. A sub-committee would have many less. A select committee could have any number, but probably no more than about 20, but probably less. I'm betting it was a select committee specifically composed to examine organized crime (a la the McClellan and Kefauver committees).

Given that the Senate (even then) would have more than ten committees, it's easy to see that a Senator would have to serve on several committees. Geary could have been a member of the that Senate special committee, but could also have been the Chair of "my own committee". If he was a visitor, then yes, he would have had to have the Chair's permission to sit in. Given that committees are established by resolution though, it's doubtful that he would have been permitted to ask questions or make comments.