Do non-human primates have culture

Yes. We used to think about certain behaviors, especially tool use, as being primarily human. However, spontaneous tool use has been observed in many species of non-human primates. Chimpanzees make "wands" to fish for termites, for example. Capuchins (the new world monkeys that most people think of when they think of a generalized monkey) also use tools--they use rocks to crack open nuts.

Of course, crows can also make and use surprisingly sophisticated tools, so we probably the main issue is that human beings like to think our behaviors are more special than they are.

Yes, primates have learned sign language, how to count, can use tools, show empathy, and a sense of humor, to name but a few behaviours that are usually thought to be human.