This use of exist instead of be is common, not only in mathematics, but also in philosophy. Although in everyday life, we make little distinction, if any, between the two, in general, the verb be when used independently, seems to carry with it a sense of sentient existence, whereas exist lacks that connotation, and is therefore more "neutral".
Now, for the word order in the sentence. As Edwin Ashworth mentioned in his comment, in everyday language we would probably find a sentence like:
Can a 7 foot spider exist?
Can there exist a 7 foot spider?
However, in the fields of philosophy and mathematics, one usually is not interested in teh existence of such mundane phenomena which can so concisely be described.
You could phrase your sentence as:
Can an uncountable planar graph exist?
However, it is not unlikely that in a similar context I am not wondering about a mere uncountable planar graph, but one that has many more properties. I might wonder a about an uncountable, limited dimensional, planar graph that is not identical to an earlier-described construct.
Now, if I ask:
Can an uncountable, limited dimensional, planar graph that is not identical to an earlier-described construct exist?
The sentence becomes a lot less clear: halfway the sentence I start wondering what that thing you are describing is supposed to be able to do!
Therefore the main verb, exist is moved to the front of the sentence, so it is clear from the start that I want to know something about the possible existence of what I am about to describe:
Can there exist an uncountable, limited dimensional, planar graph that is not identical to an earlier-described construct?