Is it permissible for my sister to have a legitimate viewing of the young man proposing to her, outside the home, without the father's knowledge, and what are the conditions if that is permissible?

It is permissible for the suitor and the fiancée to see each other so that the approval is based on clear knowledge. The suitor may look at what encourages him to marry her, such as what is commonly visible, like the face, hands, neck, and feet, provided there is no seclusion or temptation, and a mahram is present with her. Care must be taken to ensure that the father does not know about his daughter seeing the suitor if he would be angry about it; if his رضا can be obtained, that is preferable.