I have no idea what Galadriel offered Pippin (unless it was a comfortable bed, lots of food, pints). Isn't he the one who wouldn't say? I'll have to get back to the source material when I get home.
"he looked and I understood."
That sounds to me like Sauron communicated by telepathy through the palantir. Or is it that all communication through the palantir was by telepathy?
Post palantir:
Again, I'd like to have the source material on hand before I answer more fully, but I think our poor hobbit "grew up" and left his prankish ways behind.
This is the easiest of the three questions: In the general sense he understood for the first time the true measure of the darkness they were fighting (considering all he had been through up until then, that may sound amazing, but for one as unprepared as Pippin was at that time to stare into the Palantir and "link" mentally with Sauron's darkness and power that would certainly have been a revealation).
In a more personal sense, I think he probably also meant he began to understand the weight and horror of Frodo's burden. The level of the power they were dealing with on their mission became clear to him. Even the Steward of Gondor and Saruman were driven mad by wrestling with Sauron in this manner, how much more a revealation this would be to someone as innocent as Pippin at that time.
For the other two questions, I too must flee to the source material.