As per all websites or books that contain this Stotra, it was composed by Rishi Vashishta.
The Phalasruti mentions this:
Vasishtena Krutham stotram,
Sarva sampathkaram param,
Trisandhyam ya padenithyam,
Sa hi swargamavapnyuth.
This prayer composed by sage Vasishta, Is one which gives all types of wealth, And whoever chants it at dawn, noon and dusk, Would
enjoy the pleasures of heaven.
Another website (Stutimandala) also mentions the same Phalasruti given above (from Hindupedia):
Those who daily study this eulogy written by Vasiṣṭha — which
destroys all the health-related problems, which brings wealth quickly,
and which increments the lineage — indeed approach the heavens.
[Note: I have only answered one of your many questions. So, it is a partial answer. I do not know the other answers like when it was composed and what were the reasons behind the composition etc.]