Are dimensions exceeding 3 actually comceivable or are they purely intellectual constructs? Is this even debated in philosophy?

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If I understand your question correctly, it's whether there really could be more than three dimensions in physical space. The best answer, I should think, is yes. One reason is that there are serious physical theories that assume the existence of more than three spatial dimensions: string theory is the example I have in mind.

More generally, though, it's not clear why we should doubt that this is possible. The fact that we can't represent it to ourselves imaginatively doesn't seem like a very good reason. We can't represent curved space-time to ourselves imaginatively, but if general relativity is right, space-time does curve. We have a notoriously hard time representing quantum mechanical objects to ourselves imaginatively, and yet quantum mechanics is the cornerstone of much of our physics.

We can even say things about what it would be like to live in a world with more than three spatial dimensions. Consider: think of a plane in 3-space, and imagine a walled square in that plane. An object can enter the square without passing thought its walls; it enters from above or below. In a 4-dimensional space, if we picked a walled cube in a 3-dimensional "hyperplane" of the 4-space, an object could enter that cube without passing though its walls: from a region outside the plane. Those with better geometric skillz than I could offer more elaborate examples. (See also Edwin Abbot's Flatland for a delightful 19th-century exploration of this theme.)

One other thought: though we can in some sense "imagine" three dimensions, when you start trying to think about the details even of three-dimensional geometry, it's interesting just how fast imagination runs out and you're forced to adopt a more purely geometrical or mathematical approach. The limits of what we can picture or grasp intuitively aren't a very good guide to what's really possible, let alone true!

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