"Were any aspects of the anarchist movement successful?" Pretty complicated question. Of course, the anarchists didn't succeed per se, or else we'd be living in a very different society. But they did contribute to many changes in people's lives then, and helped change a lot about history's course. Anarchists were very involved in the labor movement of course, and played important roles in a lot of watershed moments. Some anarchists like Emma Goldman were early advocates of birth control, free love, gay rights, and various other causes that few others were talking about. Anarchist culture and mutual aid also played a large role in the daily life of many immigrant communities.

"How prominent was the movement?" Many anarchist periodicals had a circulation of tens of thousands (see http://katesharpleylibrary.pbworks.com/w/page/13175715/Anarchist%20newspaper%20circulation for the nitty gritty details), and these periodicals were shared and read aloud, so in total had a potential reach in the hundreds of thousands in a time when the population was much smaller than now (not to mention the reach of speeches, books, theater, etc.). In some cities among certain ethnic groups, anarchism was the most prominent force on the left. The public was generally aware of anarchists, even if many imagined a "bearded, bomb-throwing anarchist" trope.

"How much of a threat was the anarchist movement to the US (government, industry, etc.)?" The government considered anarchism a fairly significant threat, passing several anti-anarchist laws (including the crime of 'criminal anarchy' and restricting anarchists from entering the country), and eventually raiding anarchists, shutting down their presses/interrupting their mail, and deporting them en masse. The legal justification for doing this despite the right to free speech was that anarchism, even just philosophical anarchism, posed a threat to the government of the US, which had a right to defend itself. The fear of anarchist violence was a major precipitant of both the FBI and Interpol.

"How did anarchism align with other movements of the time (labor, Marxism, suffrage, civil rights, etc.)?" Very aligned with labor. Some anarchists were very involved in the IWW, others didn't think unions had much revolutionary potential. Fairly aligned with Marxism, but that gets very complicated depending on the time/organization/ethnic group (keep in mind that 'Marxism' includes both anti-parliamentary IWWs and Bolshevik Communists, and some anarchists identified with Marxism to a certain extent). The suffragism/feminism question is also complicated, anarchists believed in women's liberation but they often didn't agree with certain goals of liberal women's groups. Civil rights is also a bit complicated because of the time period and the geographic location of largely white, immigrant anarchists. They certainly saw themselves "on the side" of the oppressed, but it's not like it was Selma. The Communist Party in the 30's is more known for organizing black workers than anarchists were (though the anarchist-influenced IWW is also noted for being more racially inclusive and ethnically diverse than other unions of the 00's and 10's.)

"What were watershed moments in the movement (labor unrest, international conflict)?" Probably best to look these up on Wikipedia to start (I'd be glad to answer follow-up questions): -Haymarket affair - 1886 -Killing of President McKinley - 1901 -Bread and Roses Strike - 1912 -Paterson Silk Strike - 1913 -First Red Scare - 1919-1920 -The Trial of Sacco and Vanzetti -Spanish Revolution - 1936