Unless specifically mentioned, you don't need a dance partner to take a class. The best way to learn Salsa is to practice with different partners. Therefore, almost every teacher uses a rotation scheme to switch partners during a class. Dance schools have the responsibility to balance the number of male and females (although this is not always easy).
LA style is the dominant style in Amsterdam. You will find LA Style dancers at every party.
New York style is not very prominent here, and there are no specific parties for NY style. Some of the better dancers dance New York style (either Eddie Torres or Power 2) as well as LA style, so your best chance is at the parties with high level dancing.
Probably the parties organized by Cuban dansschools. The parties are named Fiesta Cubadores,Party Salsamoso, Salsa de Cuba and Fiesta Cubana (the last one isn't related to a dansschool).
Depends on who you ask. But check the Wikipediafor a reasonably accurate overview
I think you misunderstood: I don't give lessons myself. Check the lessons section for links to dance school sites.
My DJ skills are limited to salsa (occasionally play a merengue or reggaetonsong as variation)
My advice is to do a couple of try-outs at various schools. There are large differences in teaching style, atmosphere, and accommodation between the schools. Choose what fits you best.
Sorry, I don't know much about anything else but Salsa :-). I don't think there is much Swing dancing here - at least that would explain the high Google ranking of this site on the query 'Swing Dance Amsterdam'.. Update: you might want to check out lindyhop.nl.