Are the aforementioned actions and feelings—such as jealousy, ridiculing a friend who is engaged, and thinking about her husband marrying the questioner's sister after learning about their problems—considered envy, especially since the questioner is remorseful and prays for her friend?

Envy is wishing for the disappearance of a blessing from the envied person. So, if you did not wish for your friend's engagement to end, then you did not envy her. Mere jealousy does no harm as long as it is not accompanied by envy. As for ridicule, it is forbidden by Islamic law, whether motivated by envy or otherwise, due to Allah's saying: "O you who have believed, let not a people ridicule [another] people," and the Prophet's (peace be upon him) saying: "It is enough evil for a man to despise his Muslim brother."