Upon being reminded of his mother's recent death, Ben began sobbing, tears streaming down his face.

The correct spelling is "sobbing."

Yes, it is the present participle of the verb (to sob) and may be used as a verb form, as a gerund noun (e.g. sobbing was heard), or as an adjective (sobbing women).

I when to the toliet sobbing because everyone was teasing me as my hair was ugly.

I sobbed because I did not pass the exam.

"Early sobbing of the morn" refers to the early morning hours when there is a sense of sorrow or weeping present. It suggests a melancholic or emotional start to the day.