I don't know about your first question, although it seems safe to say that there are only finitely many characters that can be distinguished by human eyes at 1 meter. Similarly, for any particular machine we can build, I think it would only be able to distinguish between finitely many characters written, say, a cell of 1cm by 1cm. If we think of characters as closed subsets of the unit square, then the space of characters is compact in the Hausdorff metric. Therefore given any infinite set of characters, some pair of them will be "very close" in the Hausdorff metric and therefore (I think) practically indistinguishable.
The definition of a Turing machine has nothing to do with actual paper, and regardless of the answer to the first question, the state space and alphabet of a Turing machine are both finite by definition. One could generalize the definition of a Turing machine to allow uncountable state spaces and alphabets, but I don't think that the study of such generalized Turing machines would bear much resemblance to intuitive notions of "computation" for the reason stated above.