In my book of Mallarmé's poetry1, the notes for the poem say that Mallarmé is describing a stained glass window of Saint Cecilia. They point out that his original title of the poem was much more informative: it was «Sainte Cécile jouant sur l'aile d'un chérubin (chanson et image anciennes)» — in English, “Saint Cecilia Playing on a Cherub's Wing (an ancient song and image)”, as can be seen in the earliest extant draft of this poem.
In their interpretation of the poem (which I would agree with), some of your questions are answered by either their translation of the poem or their notes:
- The saint is indeed Saint Cecilia, patroness of musicians. In ancient religious paintings, she was usually depicted with or playing a viol or organ.
- The angel is one of those cherubim who are omnipresent in the background of ancient religious paintings.
- It's not a sandalwood tree; the viol Saint Cecilia is playing is made of sandalwood.
- The book is one containing the Magnificat, an ancient Christian hymn.
So what is actually going on in the third and fourth stanzas where Saint Cecilia is playing the cherub's wing? The book1 suggests that the angel is actually in the background, behind Saint Cecilia, but the way that they are depicted, his wing looks like a harp, and Saint Cecilia's finger looks like it is brushing it, so she looks like she is playing the angel's wing.
One objection to this interpretation might be that the action contained in the poem could not possibly be depicted in one stained glass window:
In the first verse, Saint Cecilia seems to be holding her viola, which is partially concealed from the viewer.
In the second verse, Saint Cecilia is laying a hymn book open (étalant) on a table or music stand, I would assume to play or sing from it or so that other people can sing from it.
In the third and fourth verses, she now has neither her viola nor her hymn book (without the old sandalwood or the old book) and is somehow using her finger to “play a cherub's wing”.
I would answer this objection by bringing up Baudelaire's poem “Une gravure fantastique” — in English, “A Fantastic Engraving”. Baudelaire similarly describes Death as performing more actions than could possibly be shown in one image. The French Symbolist poets were heavily influenced by Baudelaire's poetry, and it wouldn't be at all surprising if Mallarmé was using the same conceit as Baudelaire here.
I see that on the Internet, people have suggested another way to solve this conundrum—by postulating that there is more than one stained glass window showing Saint Cecilia. I don't like this reading as much; I prefer to think that in the window, she has just set down her ancient gilded sandalwood viola (maybe that's why it's concealed), looked up from her hymnal, and reached out a finger to strum the plumage of a passing cherub's wing.
Finally, why is Saint Cecilia a musician of silence? She's an image in a stained glass window, and so incapable of making any actual sound.
Untangling the syntax of this poem is somewhat difficult. Here is my attempt at translating it into English while conveying the syntax of the French. (See the edit history for a more literal version which doesn't have as much rhyme and meter.)
In the window which conceals
The old sandalwood, whose gilt wears thin,
Of her viol that once sparkled
With the flute or mandolin
Is the pale saint, laying open
The old book, which unfolds,
Of the Magnificat that streamed
At vespers and compline of old:
On this ostensorial pane
Brushed by a harp the angel shapes
With his flight through the evening air
For the delicate fingertip
Of the digit that, without book
Or sandalwood, she is striking
The plumaged instrument with,
Musician of silence.
I'm not a native speaker of French, so I may be wrong here, but it seems to me that there is something a little unexpected with using jadis (long ago) to modify the participles étincelant (sparkling) and ruisselant (streaming) in the first two stanzas of the poem; I would expect those to be associated with the present. If I'm right about this, I'm sure it's quite deliberate on Mallarmé's part—the action is happening both now (in the stained glass window) and also long ago (in real life).
1 Stéphane Mallarmé, Collected Poems and Other Verse, translations and notes by E.H. and A.M. Blackmore.