Yes, both of her quests are repeatable. Her 2nd quest is repeatable right away.
Repeatable quests leave your Completed section (even if they've been completed once before) until they are completed again. You restarted it, and that's why it's not showing up in your Completed section. If you forfeit it, it will eventually return to your Completed section (this is only true if you've completed it at least once).
And by "Next, Next, Next" she just meant that he saw she had a quest available, and simply clicked through its dialogue without realizing restarting it would take it out of his Completed section.
Repeatable quests leave your Completed section (even if they've been completed once before) until they are completed again. You restarted it, and that's why it's not showing up in your Completed section. If you forfeit it, it will eventually return to your Completed section (this is only true if you've completed it at least once).
And by "Next, Next, Next" she just meant that he saw she had a quest available, and simply clicked through its dialogue without realizing restarting it would take it out of his Completed section.