word choice - Which of "couldn't", "can't", and "doesn't" for 13/13/13?

The main use of would is to express ‘unreal meaning’. Secondary uses are to express habit, volition and prediction. The expression of ‘unreal meaning’ is also the principal use of could, but it can also express possibility, ability and, less often, permission. The expression of possibility and ability are the main uses of can, which also has a less important use in expressing permission. Would and could are not subjunctive forms. Do, unlike the three preceding verbs, is not a modal verb, but a primary auxiliary verb used, among other things, to form the negative of lexical verbs.

To say that 13/13/13 wouldn't exist is to suggest that it is unlikely to happen. Since we know that it won’t happen, wouldn’t is an inappropriate choice, so we should say instead 13/13/13 won’t exist or 13/13/13 doesn’t exist, although it is more likely that the thought would be expressed as something like There will be no 13/13/13.

My car can’t start is an improbable thing for anyone to say, because, unless we are being particularly whimsical, we don’t credit cars with ability in quite the same way we do with humans. We would normally turn to the negative form of another modal verb, will, and say My car won’t start. This is an instance of using will, in its negative form, to make a ‘present prediction’, that is, to say that something is likely, or, in this case, unlikely, to happen now.