There wouldn't be much difference between "they visit" and "they are visiting".
I would use "culture" because, as a general rule, a country only has one culture. I don't mean to say that many countries aren't multi-cultural, of course they are, but most countries have, for example, one generally accepted standard of dress, or of speaking to strangers, or how much children are seen and heard. There are plenty of other examples of a country's culture.
I would find "the cultures of the countries" somewhat unnatural.