Can A Strong Blow To Back Cause Herniated Disc?
Those episodes are unlikely to be the cause.
Detailed Answer:
I read your question carefully and I am sorry about the troubles you are going through with your back.
Regarding your specific question though, I have to say no, those pushes shouldn't have had any effect on developing your herniated disc, human spine does endure much more than that during its lifetime. It is very rare that a strong blow, a big fall or some other accident causes a disk herniation and those episodes you describe do not appear to be that injuring physically.
Disc herniation is usually a gradual summing up of the effects of a lifetime of wear and tear on your spine combined with the intervertebral disk losing water content. Some degenerative changes of this kind happen to all of us with age, in some individuals it is more advanced due to genetic predisposition as well as other predisposing factors such as obesity or demanding physical work.
Apart from medication physical therapy with stretching exercises, or traction therapy is equally if not more effective in relieving your symptoms, so if your haven't already you should try it. I suppose you have already tried over the counter painkillers as well as neuropathic pain drugs such as anticonvulsants ( pregabalin, gabapentin) or antidepressants (duloxetin, amitryptilin)
Surgery is usually as a last resort in case those measures do not work.
I remain at your disposal for further questions.