She got over me within weeks!

I dated someone for four years and it ended because we felt as if there were too many things we couldn’t resolve. I still have strong feelings for her, but I knew the relationship wouldn’t last and had no choice. I recently found out that she has been with someone for a few months now, which means she began dating just a few weeks after our break-up. I haven’t been able to cope with this because I can’t understand how you can spend four years with someone and then just get over that person in a matter of weeks. Does it mean our relationship meant nothing to her?
You can’t speak for someone else, nor can you assume what your ex-girlfriend must have felt when your relationship ended. Everyone has different ways of coping, and, for some people, another relationship is sometimes the only way of moving forward. To assume the four years preceding that meant nothing to her is a stretch, and unfair given that you are presuming things without any facts to back you. Why not consider why your relationship ended, accept that she is no longer with you and has a right to make any decisions about her life, and work on your own grief so you can heal and start to move on too?

My ex-boyfriend has been trying to get in touch with me, but I don’t know if this is a good idea. He says this isn’t about getting back, and just wants to stay in touch but I don’t know if this is good for my own mental health. Should I allow him to chat with me now and then?
That is your call entirely because you alone can decide if chatting with him has any positive or negative effect on your well-being. If you aren’t interested in staying in touch, you have the right not to. If you want to give it time before you begin speaking with him again, that is your choice too. Either way, you control this.