What is the ruling of the Shari'ah on getting to know a girl with the intention of marriage and proposing to her after a financial hardship has ended, with the knowledge of her family, and is it permissible to offer her a gift in the presence of her friend and with the knowledge of her family?

If you express your desire to propose to a girl according to Islamic guidelines, there is no harm. However, communicating with her for the purpose of getting acquainted is fraught with risks, as she is a non-mahram to you, and this could be a pretext for temptation. A fiancée, like any other non-mahram woman, is forbidden to be spoken to or to be alone with, except for a legitimate necessity. The best way to learn about her is to ask trustworthy individuals who know her. There is no harm in giving her a gift, as long as there is no suspicion or fear of temptation, and the gift can be delivered without direct contact. If you are not ready for marriage soon, then it is better not to rush into a proposal.