What does it mean when someone has 'issues' with something?

The OED has a draft addition:

In pl. orig. and chiefly U.S. Emotional or psychological difficulties (freq. with modifying word); points of emotional conflict.

If it doesn't have the modifying word mentioned there, it would often be assume that psychological or emotional was elided. This was made popular first among circles in which psychoanalysis was popular, and then among circles where pop-psychology was popular (which at least is cheaper than psychoanalysis, if nothing else).

Of the three hypothetical uses you give:

The issue with your clock could be seen as a facetious use, which is a common enough use (hinting at a certain scorn for those who use the term in terms of psychology, though that could include self-satire).

The issue with the car could be seen as a facetious use that also used humorous understatement (I'd expect an unqualified issue to mean perhaps gestured insults after they cut you off).

The issue with the heart we'd take as eliding medical rather than psychological.

I'll stick with my comment that, as a starting point for discussion, an issue is, in one definition, either a vital matter or an unsettled matter. (See here, at 6b for example.)

Also consider that people are naturally prone (in my U.S. English experience, anyway) to use exaggeration for effect, either to sound humorous, to be emphatic about a point, to communicate in a thought provoking way, or to provide metaphoric imagery.

That being said, I would say that your example "I will have issues with the clock" veers in the direction of exaggeration, stating that something of vital importance (as in urgent, pressing, compelling) is associated with the clock - there is a meeting scheduled. This fits perfectly within the inclination to use a figure of speech for any of the purposes I mentioned above.

When Suzannah has issues with self-confidence, she has an unsettled or unresolved matter concerning her self-confidence. There is a problem, question, or dispute about the state of her self-confidence.

If you say you have an issue with another driver, then someone would infer that there is an unsettled matter (a problem, question, or dispute) between you and that driver.

An issue with one's heart is a vital matter in the most literal sense, but to say there is an issue with one's heart is also to say there is an unsettled matter with the heart, just as above.

None of these examples are even a far stretch from the cited definition of the word issue. As to how far you can go, the limit is determined by the point where you cease to be understood. English users love to engage in figures of speech and will even invent their own at times (and there's no reason not to, aside from reaching the point of being incomprehensible).