Why would someone who is already in pain wish to inflict it upon others?

Someone in pain probably feels a lot of anger and hurt. Sometimes people with anger and hurt can misdirect their feelings to other people. There are various reasons why an angry and hurt individual might want to project their feelings on to others such as attention, projection, or an outlet. There are many more other reasons.

Because a person who is in pain , will feel rejected and unwanted so they push away others ,and they will start feeling the pain

Their mindset might be that, if they are in pain, everybody else must be too. Whether it's mental or physical, they feel as if they don't deserve this pain, so they want others to be unhappy as well.

because they want someone to understand how they feel and feel it with them. They dont want to be alone and feel hatred againt those who arent feeling what they are feeling (sometimes)

A lot of the time, people who are in pain tend to wish others feel pain too. This is because they're incredibly angry at the pain they suffer and so they believe it's only fair that everyone else feels pain with them. However this isn't a healthy emotional expression, it's much healthier to help others and seek help rather than upset everyone else too.

The problem with people in pain is simply either they do not realize they are hurting others, or they believe this is some kind of revenge. I suppose sub consciously, they also redirect their own pain to others as a way to express themselves in both cases. Either way, this behavior must be stopped, and the person must be made aware of how they are hurting others and giving alternatives to find peace and heal themselves before it is too late.

Because they want the others to know what they are going through, it somehow makes them feel less pain because the other people are feeling far more pain than the bully him/herself. I do not understand completely, but I believe this is one of the biggest reasons.

We live in a world of good and evil, with many gradations in between. The coarse, lower substance of evil is a distortion of the divine. Without the divine it cannot exist. Nonetheless, it derives merits out of bringing beings to act out our base, bodily selves. The pleasure that we get from diminishing others can get addictive, if we do not realize the full cost of doing so.

So that they aren't alone and someone else goes thought the misery they go through on a daily basis.

The reason that may be is because misery loves company. So if your in pain the most used response is to have others in pain as well as a self gratifying result