What exactly causes the normal tight and aching chest/throat/face when crying? Because I get similar pain at other times, sometimes exacerbated by deep breaths. Clear chest ct, neg cardiac work-up.

Muscle and low CO2. Crying usually consists of many muscles tightening, sometimes in almost spasmodic nature. Combine this with associated hyperventilation, and chest tightness, a sense of inadequate breath, and perhaps even tingling can occur. It simply depends on how violent the crying is.
Nerve irritations. We have sensory nerves in our airways, trachea etc that let us know when things are not right such as bronchitis with its irritation and coughing. Crying and deep breaths may trigger the same receptors with similar tightness/aching. When bothered, try to stop the hard breathing by relaxing and breathing slowly in through the nose, hold it briefly and then gently exhale with a slow sigh. No smoke!