Is it permissible to look at a woman one intends to propose to, and at what is visible of her (such as the face and hands), with the aim of encouraging the suitor to propose to her, before she or her family are aware, and with the intention of proceeding with the proposal?

It is permissible to look at one's fiancée without her knowledge or without requiring her knowledge. It is permissible to look at her face, hair, and hands in the presence of a mahram, and to look at her approaching and departing, based on the Prophet's (peace be upon him) saying: "Look at her." This is also supported by the action of Jabir (may Allah be pleased with him) who used to hide to look at the woman he proposed to.