I do not think Darcy and the Colonel really have business at Rosings. I think the “business” Col Fitzwilliam talks about is the traveling arrangements. He is dependent on Darcy because they probably traveled in his carriage with his horses. IMO Darcy has put off leaving because of Lizzy—he just cannot drag himself away! (;D) Darcy and the Colonel visit at Easter; I think of their visit as an annual respect/courtesy visit to their aunt and niece. The annual tour of the park is, IMO, kind of courtesy to their aunt—they walk around the park and look at stuff to ensure everything is in order because there is no protecting male, “man of the house.” I think this is more or less a courtesy because despite her flaws Lady Catherine seems capable of running the park.

Colonel Fitzwilliam seemed really glad to see them; anything was a welcome relief to him at Rosings; and Mrs. Collins's pretty friend had moreover caught his fancy very much. (Chapter 31)

Col Fitzwilliam fancy’s Lizzy but I do not think he intended to show her too much attention since he could have no honorable reason for doing so. He is prevented from marrying where he chooses because he does not have enough money to do so. I think Helen Halstead may be right about his wealth; I suspect he is not a pauper but has not enough money to support a wife in the style to which he is accustomed. It is hard to say and I am just speculating—the Colonels father is an earl and Lady Catherine’s brother so the Colonel may also be expected to marry very well. I think he got carried away or he thought by Lizzy’s manner that he had somehow misled her or both. That is why I have been interested in how Lizzy’s manner affects him—is it him, her or both. When the Colonel says “he speaks feelingly” I too think he is saying he speaks from personal experience but I agree with Lizzy that the Colonel probably really has not known of self-denial and dependence. ;D