The Buddha did indeed believe in past lives and he remembered all of his. Zen Buddhism is one of the many schools of Buddhism. It believes all that the Buddha taught. Its main emphasis is on meditation.
As Buddhism has spread to different countries, it has always been very flexible. Every culture, every country, every civilization has different needs, and Buddhism has always adapted to those needs.
Take, for example, that debate over anatman. The Hindu believed that atman was a divine spark, completely perfect, that went from one body to another in reincarnation. The atman, as they believed (and possibly still do, I do not know), never changed because it was completely perfect. It was like a diamond set in a metal ring. When the metal of the ring wore out, the diamond was set in a new ring, and when that wore out in yet a new ring, and so on, and so on. The diamond never changed, the rings did. So, in their idea of reincarnation a perfect soul is in a body and when the body wears out and dies, it then is reincarnated in a different body, but the soul itself never changes. The Buddha disagreed with that. The Theravada Buddhists who live in the Indic countries emphasize the idea of anatman very strongly because their Hindu neighbors believed and perhaps still do believe in this immutable soul.
In the West, the concept of an immutable soul does not exist. For one, in the early history of the Western civilization Heraclitus said that everything flows, everything changes, nothing stays the same. This is an important part of the Western philosophical tradition. Everything changes. A Latin proverb says
tempora mutantur et nos mutamur in illis, that is,
times change and we change in them.
Now, when Theravada Buddhists came to the West, they continued to emphasize the teaching of anatman. And to many Westerners that sounded like they were saying that we did not have a soul. What they actually were/are saying is that we do not have an immutable soul. And a proper reaction for any Westerner should be:
"What's an immutable soul?" And then we get to a rather paradoxical situation where the Theravada teachers introduce Westerners to an idea that is totally foreign to them (the immutable soul) and when the Westerners finallty get it, the teachers then say that the idea is false. It would be like if some Eskimos moved to Central Africa and started teaching the locals how to preserve heat in the winter. The Africans would not know what winters is for starters and would most likely not quite get it. Similarly, many Westerners do not quite get the teaching of anatman and misinterpret it as if the Buddha had taught that we have no soul.
Now, Zen Buddhism comes from China. It is actually called Ch'an there but the Westerners learned about it from the Japanese who learned about it from the Chinese. And in Japanese they use the same Chinese character for it but they pronounce it Zen. Additionally, Zen also spread from China to Korea. None of those cultures are Indic. They've had Buddhism for many, many centuries. And they are not terribly concerned with explaining to their students the differences between Hinduism and Buddhism. After all, there are no Hindus in China, Korea, or Japan (except for tourists and business people from India, of course). Of course, the Zen monks are aware of the doctrine of anatman, but it is just not terribly important to them.
Buddhism is relatively new to the West. And it has come here from different sources. So, particularly in the US, which has large immigrant communities from different parts of the world, you have temples and teachers of all schools of Buddhism. Their main goal was to cater to the immigrants from their own countries but as Buddhists are always willing to explain Buddhism to those who ask, they have also been teaching different Westerners. And as a result, many Westerners are completely confused. You will find Westerners who claim with authority that Buddhism teaches this and Buddhism teaches that, and many of them have no clue what they are talking about.
And, of course, there also are Westerners who do understand Buddhism. Eventually Buddhism will adapt to the West and a uniquely Western school of Buddhism will develop. I had a teacher who is an American and who studied under a Chinese master and was ordained monk by a Japanese master. His students were almost exclusively Westerners, mostly Americans (but I was his student, too, and I am grew up in Europe). He never talked about anatman to his students. I had learned about it from Theravada Buddhist monks in Washington, DC, and yes I had to first learn the totally strange idea of an immutable soul, just to be told that the idea is false.

But my American teacher told us such things as
"There is salvation but there is no Savior." The Sri-Lankan Theravada monks never said any such thing because to them the concept of a Savior is totally foreign. Yet, we here in the US are constantly bombarded by people telling us to accept Jesus as our personal Savior. It does not matter what your religion is (or even if you don't have one), we are still bombarded by it, and to me the concept of salvation without a savior was much more enlightening than the falsity of the concept of an immutable soul.