How closely related are laughing and crying?

I think since they are both involuntary responses to a stimulus that in this respect they are related. The laughing at funerals could be from something genuinely funny happening but mostly it is probably an unconscious manifestation in reaction to stress and anxiety.
The ability to rapidly shift between the two extremes is also an emotional reaction when ones emotions are all over the map. Seems about right to me.

This article from the Atlantic sheds some light on it.

I simultaneously laughed and cried when Porsche announced they were producing an SUV.

One easily flows into the other.

For many, they are separated by one month’s salary.

I remember laughing at my fathers funeral and my husbands grandmothers funeral. Several people actually at both. It’s common to remember the best of times and share them with others at funerals. So you tell tales of the deceased and often its the funniest times you remember best. So you can laugh at what happen and then cry as it starts to sink further in that there will be no more future of tales and that a life that bought laughter will not longer do so. But you tell the tales because you want those who may not have had that experience to know what will be missed. Joy and sadness walk hand in hand. You are happy to see your children grow into adults but sad to see them no longer need you and to be able to have them cuddle in your lap as they once did. You are happy to see them marry but sad that the marriage will be their first priority family. It’s why people cry at weddings. With every year they grow further away, but in the end, you prefer that then them only having you till you die and they are left alone. You want their happiness. Which will bring you joy and pain.
Yin and yang. It’s everywhere.