To what extent do Zen students study Dhamma?
I found my way into Buddhism via Zen since the teacher 'who appeared when needed' was a Zen and Taoist practitioner. This was fantastically lucky since Zen cuts straight to the chase and I'm impatient with details. As a lazy person 'just sitting' was right up my street. 

I started by reading 'Cultivating the Empty Field', a collection of poetry by Master Hongzhi. The preface and introduction are brilliant and from a philosophical perspective explained all I wanted to know. At the time these were by far the two best and most useful philosophical pieces of writing I had ever read. 

The dharma in all its details is not especially important since Zen goes straight the heart of practice. But the situation is deceptive. The teachings and practices of Zen are grounded in the metaphysics of Nagarjuna. Thus Zen is iconoclastic as is Mahayana. Nothing really exists or ever really happens and all our mundane idea have to be knocked-down.  

There is very little need for organised teaching. I was advised to sit down and give up everything. That's it really, the core of the method, supported by pondering on koans, the occasional touch of austerities and so on.       

I would recommend the aforementioned book as a way into this topic. I have no patience with the 8 levels of this, the 43 levels of that, the 9 stages of the other and so on for to me they are merely didactic devices, albeit valuable where they are helpful. It won't suit everyone, but some prefer to burn their books and leap-frog all that. 

I feel Zen is the best approach for anyone coming from scholastic philosophy since it is free of clutter and has an explicit, complete and well-formed metaphysical scheme. Whether it is best for you only you can know. It combines well with other simple-minded approaches like philosophical Taoism (not the later religious Taoism) and Sri Ramana's method of self-enquiry.          

You ask what is Zen doctrine. It is the Mahayana doctrine stripped-down to its most-minimal practical essentials. One sits and waits for the dust to settle, and as it does ones realisation grows and life improves. No need to fight tendencies and attachments like a battling saint for they gradually fall away in the silence and peace. Also, koans are an effective method of unsettling ones usual way of thinking and getting to grips intellectually and 'intuitively' with non-duality.

Hope this helps. For more of an answer the question would have to be more specific.