No, Jane and Elizabeth did not send Lydia money after she married Wickham. It was Mr. Darcy who provided financial assistance to ensure Lydia and Wickham's marriage could take place and to settle their debts.
Lydia was supposed to keep Darcy's involvement in her wedding secret. Darcy had provided what was, in essence, Lydia's dowry, the money Wickham demanded to marry Lydia (though Lydia might have been unaware of the fact that Wickham would not have married her otherwise). This money was was used to buy Wickham an army commission and pay his debts. Darcy paid for the wedding, and attended it as best man.
Mr.Darcy is actually from Pemberly which we learn is near Derbyshire. His hoiuse or residence is not exactly in Derbyshire but a little on the aside. In the later part of the novel ,Elizabeth goes with her aunt and uncle to the lakes but cant make it there so instead they check out Derbyshire and hence Darcys house
Mr. Darcy gave George Wickham the money to marry Lydia in order to save the Bennet family's reputation and to ensure Lydia's well-being. Wickham had previously tried to elope with Lydia without intention of marrying her, which would have brought shame upon the family.
Wickham eloped with Lydia and was found living with her, unmarried and with no intention of being married, in London. He is bribed to marry Lydia to save the reputation of her family. We do not know the full extent of the bribe, but it included a commission in the regular army, having all his debts paid, and some additional money.
Mrs. Gardiner reveals that Mr. Darcy was instrumental in arranging the marriage between Lydia and Wickham by paying off Wickham's debts and securing a comfortable living situation for the couple. She explains that Mr. Darcy did this out of his affection for Elizabeth and to protect her family's reputation.
Darcy makes Wickham agree to marry Lydia. He repays a large chunk of Wickham's debts but arranges for Mr. Bennett to pay the debts to the shopkeepers in Meryton. Darcy also pays someone money so that Wickham can get out of the volunteer militia regiment he had joined that put him in contact with the Bennett sisters. He then enters the regular army and is stationed in the north of England, away from Hertfordshire. Finally, because Wickham has no annual income and no money Lydia to inherit should he die before her, Mr. Bennett has to agree that Lydia will still get the 1/5 of Mr. Bennett's estate after his death. Mr. Bennett is also asked to provide her an annual personal income of 100 pounds/year (a tiny amount, really) for her living expenses.
Mr. Gardiner sends a note to Mr. Bennet saying that Lydia and Wickham have been found. They are not married, and they (or perhaps just Wickham) have not intention of being married, but they will marry if Mr. Bennet provides them with a hundred pounds per year, while he lives, and sees to it Lydia gets her share of the 5000 pounds left Mrs Bennet and the daughters when he dies.
It is clear that Wickham is a fortune hunter, a man who is out to marry a wealthy woman for her money. In fact Wickham needs money badly just to cover his debts. Lydia has no money, but Wickham is marrying her. Clearly someone is providing Wickham with money, and Mr. Gardiner is the logical person to do so. At that time, the Napoleonic Wars were under way. A quarter to a third of all the young men in England were going off to war, and a large number of them would never return. Women from good families could not get jobs except as governesses. If they did not have money, they had to marry well or become dependant or impoverished. There was a lot of competition for husbands. Wickham had nearly got away with marrying Georgiana Darcy and getting her 30,000 pounds. He had also nearly married Mary King with her 10,000 pounds. If he kept his act up, running away from his debts, he was very likely to find a wife with a similar amount of money, so a marriage to Lydia required such an amount. Mr. Bennet knew that Mr. Gardiner had a family interest in seeing Lydia married and further scandal prevented. He also knew that Mr. Gardiner had a lot of money. He had no idea that Darcy would have any reason to provide for Lydia.
Mrs. Gardiner is concerned that Elizabeth might fall in love with Wickham because neither has any money, and neither have any great inheritance coming. The result of this is if they married, they would have to live on Wickham's income from the army, which would probably not be very great. The result would be that they would have to live much more frugally than Elizabeth had ever done.
Mr. Wickham tried to elope with Mr. Darcy's 15 year old sister. Mr. Wickham also squandered away the money he had received from the will of Mr. Darcy's father and he created a plot to marry Georgiana to get her money.
Darcy and Wickham had history. They had been raised together, and Darcy learned to distrust Wickham because of his bad habits, such as gambling. Wickham was to inherit a position on Darcy's estate, but asked to take money instead, which he was given. After he gambled this away, he came back for more, which was refused. He then tried to elope with Darcy's sister, and almost succeeded. In the book, they clearly do not want to be near each other. Wickham tells lies to Elizabeth and others about their connection, and Elizabeth takes sides without close examination of the truth. Later, when Darcy has reason to defend his honor on the subject, he sets the record straight, convincing Elizabeth of the truth. When Wickham elopes with Elizabeth's sister, Lydia, Darcy hunts them down and bribes Wickham to marry her for the sake of the honor of the family, to ease Elizabeth's pain on the subject. This is expensive, as it means purchasing an army commission, paying off Wickham's debts, and providing some amount of cash.
It is important for the Bennet family because Mr. Wickham marrying Lydia would restore her reputation and prevent scandal. Additionally, it would secure Lydia's future by providing her with a respectable marriage and financial stability.