How and where is it used? Will we be taught any in the course?
This course does not address the literary register of Welsh at all.
As the term implies, it is a formal variety of Welsh which is confined to writing and rarely used in spoken form. A modern form of formal literary Welsh has been used in the 1988/2004 translations of the Bible - you can find them on the web. If you look up 'literary Welsh' on the web you will find explanations of its structure and comparisons with less formal registers.
There are various standards of literary Welsh.
eg I like coffee = "Dw i'n hoffi coffi" (spoken Welsh)
"Rydw i'n hoffi coffi" (School written Welsh and also some spoken forms)
"Rwyf yn hoffi coffi" (More formal written Welsh for books and magazines - dropping the pronoun)
Yr ydwyf yn hoffi coffi (Very formal written Welsh found in the Bible and legal documents)
The course 'teaches' the first form, the spoken Welsh form, but usually accepts answers written in the first three forms.
The first two lines of the Lord's prayer are a good example of very formal Welsh:-
"Ein Tad, yr hwn wyt yn y nefoedd Sancteiddier dy enw"
Would you consider it to be useful for one (AKA me) to learn these other forms of Welsh in the future in order to achieve fluency? It is my lifetime goal to become fluent in Welsh and I'm not sure of any resources to help me learn other forms of the most beautiful language in the world.
A good grammar book such as 'Welsh Rules' (by Heini Gruffudd) will introduce more formal Welsh as it is used in modern novels and official writing. 'Welsh Rules' concentrates on the usual colloquial register of Welsh, so the more formal forms can be seen in context.
Items on news and current affairs websites such as the BBC's Cymru Fyw and Golwg's Golwg360 are usually well-written in a common formal register.
Once you have dealt with modern formal Welsh, the more old-fashioned forms are then straightforward enough to learn to read if you need to.
At the other end of the formality scale are the dialects and also slangy Welsh - in many ways these are harder to learn to understand unless you bump into them day to day.
You'll find the other forms in written blogs/websites/magazines/books etc and of course the bible.
After the Duolingo course it would be good idea to have a go at some of the written content available suitable for learners, although there aren't many free ones.