Women composers were far and few in comparison to their male counterparts at the turn of the century. Despite some prominent European female composers in the middle of the nineteenth century--namely Clara Wieck Schumann (wife of Robert Schumann) 1819-1896 and Fanny Mendelssohn Hensel (sister of Felix Mendelssohn) 1805-1847--women were still regarded as not having enough "genius" to be a composer.

As incredibly sexist as that sounds today, it was "common knowledge" back then. For instance, here's a quote from The Musical Times in February of 1887:

That no woman has ever been a great composer is an accepted fact; that she is never likely to become so, more than a probability. In these days, when woman has invaded so many fields hitherto occupied solely by men, one would think that the creative musician felt that he had a certain right to hold his head high among his brother lords of the creation...

But why is there "none to dispute" the right? Why should the creative faculty, displayed by women in other branches of art--poetry, sculpture, painting, and most of all, fiction--be denied to them in music only?

The unnamed author continues on about how women are too emotional and don't contain the faculty to compose (especially abstract music). So even in 1887, there was a stigma that women just didn't "have it in them" to be great composers. Most of the time, they were performers (very often pianists). They were allowed to tour when they were younger. Notable, Wolfgang Amadeus Mozart's sister Maria Anna "Nannerl" was seen also as a musical prodigy. Schumann and Hensel were both quite gifted pianists themselves. (Clara Schumann was actually a piano student of Robert Schumann). If women performers were to compose, it was often pieces that they could play/showcase themselves (piano solos, songs, etc.) These pieces might be published (Robert surprised Clara by publishing some of her works for their first anniversary), but they had a much more difficult road to get recognition than male counterparts. And thus, the prevailing opinion that women couldn't write "good" abstract music stuck for a very long time.

So what else could women do? Clara actually edited Robert's music toward the end of his life (and after he died). She also was one of the notable piano faculty at the relatively new Hoch Conservatory in Frankfurt. But her case was the exception rather than the rule. Most women were expected to stop performing and settle down once they were married. (This is the unfortunate reason Clara's catalogue is unfortunately short.)

So why this lengthy prelude to the question? Because it has a lot to do with probably the best example of a turn-of-the-century woman composer, Amy Beach. She was raised in a Shaker household and consequently had a fairly limited musical training (on purpose) even though she possessed an incredible talent at a very young age (although linked to Wikipedia, the cited book by Adrienne Fried Block is an excellent source!). She ended up becoming a very well respected pianist who was composing her own works despite never receiving any education whatsoever in composition.

What is incredible is that Beach destroyed some of these social stigmas, while simultaneously falling under them as well. She married a well-off doctor who was much older than her and consequently her performing career came to a screeching halt. Most of her performances at this stage were small private gatherings compared to performing for large public audiences.

She, however, turned her focus to composition where she started to defy the norms of women at the time. One of her earliest works was a Mass (a very large endeavor). She would end up composing a symphony, a piano concerto, a violin concert, and an incredible massive piano work (Variations on a Balkan Themes) that were all highly unusual for a woman composer. And the most amazing thing is that she did this without a single ounce of compositional training. Notably after Beach's symphony premiered, fellow New England composer George Whitefield Chadwick said that Beach was now "one of the boys" (referring to the other composers in the so-called Second New England School.)

It wasn't until her husband died that she was able to travel to Germany to study. But by that point, Beach had pretty much hit the pinnacle of her career. As WWI came around, Germanic styled classical music fell far out of favor.

As far as conducting, women were not allowed to conduct. This was the case through the twentieth century. As noted in a May 1984 issue of the Music Educators Journal only 3 of 218 conductors listed in the Musical America Directory in 1971 were women.

Anyway--hopefully this answers most of the questions here. I highly recommend reading Amy Beach, Passionate Victorian by Adrienne Fried Block for an incredibly detailed account of Beach's entire life. Block also wrote a book Women in American Music worth checking out. There is also a collected set of essays Women and Music that gives a good summary of the broader movement throughout time.