]Did Lizzy expect Miss Darcy on the day of her arriving at Pemberley, or the day after? And, whose arrival at Lambton is she talking about here?

I base my interpretation on the information given by Mrs. Reynolds (essentially identical to P&P2):

...Mrs. Reynolds replied, that he was, adding, "But we expect him tomorrow, with a large party of friends."

Let's assume Elizabeth and the Gardiners arrive on Day 1. So Darcy and his party are expected on Day 2. But he gets home early, on Day 1. Georgianna arrives as scheduled on Day 2.

ELIZABETH had settled it that Mr. Darcy would bring his sister to visit her the very day after her reaching Pemberley; and was consequently resolved not to be out of sight of the inn the whole of that morning. But her conclusion was false; for on the very morning after their own arrival at Lambton, these visitors came.

First part of the sentence says Elizabeth expected Darcy and Georgianna on Day 3, one day after Georgianna arrives. But they come to visit one day earlier, on Day 2 - "the morning after their (i.e. Elizabeth and Gardiners') own arrival at Lambton."

This implies Darcy's eagerness to meet Elizabeth again. Took his sister to Lambton, pretty much trundled poor Georgianna right from one carriage to another! Can there be any doubt of his intentions? ;)

] Does this mean that Lizzy realizes that she has fallen in love with him, or that she could fall in love with him?

"Could", I believe, is the operative word. Where she previously could NOT, now she COULD and this is her change of sentiment. By this time it is my opinion that she's already somewhat smitten and conflicted, but then I am one of those crazy people who thinks Elizabeth is attracted to Darcy from the moment of their meeting.