[If Lady C gives the Collinses the carriage for their ride home, why doesn't she send the carriage for them in the first place?]

Probably it would be getting dark during the return journey. She bestows the favour condescendingly but doesn't want them assuming she will have them fetched as well. Harnessing up four horses is a lot more trouble than bringing round the chauffeur driven limo. And it's not very far to walk. But she can send them home once they have performed for her - it's all part of her power game.

The Collins' gig: I think it might have been seen by lady C as a move "above their station" if they used their own transport for such a small distance. Only the wealthy kept their carriage horses ready "on tap" - Mr Bennet's, if you remember, were often wanted "about the farm".

[If the Lucases and Lizzy arrived in London "by noon" how could "the morning [pass away] in bustle and shopping"]
The morning was the period before dinner, which would be eaten at any time from four onwards - later in fashionable circles.