Octorox said:
I felt it was a bit abrupt that Padme and Anakin just sneak in through a side entrance and in the next scene they’re at the table with Dooku.

I’ll add that in the preceding scene Padme says to Anakin something like, “Whatever happens, follow my lead. As a member of the Senate, maybe I can find a diplomatic solution to this.” So the scene where Padme sits down with Dooku does follow logically. Their plan is fairly clear, and I don’t think they believe they are sneaking in the side door.

Why doesn’t Anakin just pull his lightsaber? Well, even though Anakin is naturally very impulsive, perhaps his Jedi instincts have kicked in and are telling him this isn’t the right time and to be patient. After all, he essentially does the equivalent in the droid factory scene when Jango points his blaster at him and Anakin immediately backs down (I’d actually contend this is even more bizarre).