How can I summarize and analyze every stanza of Auden's poem "In Memory of W.B. Yeats"? Who are the "rich women" and the "dogs"?

It sounds like you're writing an essay on this poem, which is a lament, or elegy, by W.H. Auden for the Irish poet W.B. Yeats. It's a big poem, but I'll try to get you started on your answer, and make sure you understand the parts you're unclear on.
The poem is, of course, divided into stanzas, but it is also broken into three sections, which is a more useful way to look at it. Traditionally, all elegies are in three sections: a lament for the dead; a section praising the dead; and a section consoling the reader. Auden's poem follows this structure.
Part I ("he disappeared in the dead of winter") uses pathetic fallacy to demonstrate how all of nature mourned the "dark cold day" of Yeats's death. The cold and the snow that lay on earth represent the feeling of coldness that came over all as a reaction to Yeat's death.
There is an element of consolation creeping into this section—"the wolves ran on through the evergreen forests"—suggesting that not the entire world was affected by this one death. But Auden refocuses the reader on the poem's subject: "for him it was his last afternoon as himself." He uses metaphor to depict Yeats as a city whose "provinces" revolted and "squares became empty." As he died, "silence invaded the suburbs" and the city of the poet was emptied.
Instead of being contained as himself, then, Yeats is now "scattered among a hundred cities," seemingly a suggestion that the constitutent parts of his soul have been divided up among others. He "became his admirers" in that all he believed lives on only in them, now, not in him. Auden uses anaphora to enumerate the many who will "think of this day" in the future as a sad day, the repeated "and" serving to demonstrate just how many lament Yeats's death.
Part II is notably brief. It describes Yeats succinctly as "silly like us," and emphasizes the key point that "your gift survived it all." When Auden mentions "the parish of rich women" here, he is referring to the upper class women (grouped as a "parish") who misinterpreted or misunderstood Yeats, particularly those in "mad Ireland" who could not understand Yeats's position on the Irish question. Yeats's poetry, Auden says in this stanza, was caused by his beliefs, by his "hurt," and will survive him; but it is not poetry that makes things happen, but the things that happen to us which make poetry happen.
In Part III, the consolation, the structure and meter echoes a funeral eulogy or memorial poem on a gravestone: "William Yeats is laid to rest." As a consoling section, this one is interesting because it emphasizes the "nightmare of the dark" in which Europe is lying, "sequestered in its hate." By "the dogs of Europe," Auden is referring to the warring nations who, at the time of writing (1940) were launching into World War Two. Where Auden finds consolation is in the plea that the "poet" will still "with your unconstraining voice" persuade us that it is still possible to "rejoice." Though Yeats may be dead, and "human unsuccess" rife, Auden prays that the "healing fountain" will start up in the current state of chaos, and memory of Yeats's poetry will "teach the free man how to praise."
I think I've covered all the lines you've asked about specifically, although my copy of the poem isn't numbered. Hopefully this gives you a good starting point.