Divorce is so difficult when both people are in agreement that it needs to happen, so I am so sorry you are having to go through this. I do not think that your (possibly soon to be ex) husband can remarry until a LEGAL divorce has taken place (which if you have not drawn up or received divorce papers, hired a lawyer and secured a court date for your hearing, it has NOT taken place). I am assuming you are in the United States, and I realize (since you said you are from Pakistan), that if you are still in Pakistan that your laws may be completely different, so I apologize if my assumption is incorrect about you being in the US).
However, the first issue I saw is that the man you married was 10 years older than you in the first place. This is not always the case, but some men really have a penchant for younger women and feel they should "trade in" for a "younger model" after awhile. My dad was remarried 4 times (younger each time) after he and my mom got divorced and I was always bothered by that, but I know that is how some men think.
But please know this man's decision to abandon his family is NOT YOUR FAULT!!! He sounds like an egocentric narcissist who is only concerned about his own needs and wants. People that are like that try to hurl insults at others and blame other people for what really are THEIR shortcomings. So, first of all, PLEASE do not take on his lies as your realities. You definitely do not deserve that.
I can tell you are an amazing mom, because you mentioned throughout your post how concerned you are for your daughter. THAT will be what gets you through!!! Having someone to focus your love and care and attention on is a true gift through the process of divorce.
Now, that being said, it is a journey and you will have good days and bad days for quite some time.
First, you can't grieve your divorce effectively until it's actually been completed. I was separated for about 3 years before my divorce was finalized and then it was another 2 and 1/2 years before our house sold (neither of us lived there the last year, it was vacant) but then I eventually had a "clean" break from my ex. I know that "in between" time is often uncertain and can be very challenging. Like you said, "he divorced you verbally so, you are divorced in your culture". If that's true, then perhaps you can consider this a "positive" and allow yourself to begin the grieving process.
Part of my grieving process was choosing to see the FREEDOM that my divorce actually brought me. I listed many of the hurtful things my ex used to do on one side of a paper and several of the positive things I was going to miss about my marriage on another side of the paper. Most of the "positive" things were related to "being married", "having a friend/companion" or something like "always having someone to talk to." When I reviewed the list, most of the things I really valued in my marriage were not connected to my "ex" personally and were all things that I could one day find again, if and when I chose to pursue another relationship. I went through the list very frequently and said, "Today I am blessed because I don't have to be deal with (I'd name a problem from the list that I was freed from when I got divorced) and while I am missing (I'd pick something I was grieving from my marriage), I believe I can move forward and become stronger for me and my children by (I'd choose a positive action I would take to try to help myself feel more positive…things like 'listening to a song and let the beautiful lyrics be about me" or "watch a favorite movie and really enjoy it etc.).
I think the pain lasts as long as you choose to hold onto it. My grandparents got divorced after 30 years of marriage and my grandma lived for an additional 20 years after that. She was one of the most bitter people I know because she always blamed my grandpa for her misery and she always hoped that he would somehow return to her and they would be reunited. The main thing I learned from watching her do that was to "let go" of something that is "dead" or else you let that "dead" thing take over you!
Don't get me wrong, you are allowed to grieve and be upset and hurt by this. But the sooner you accept it and make a plan to recover from it, the better off you will be.
Finally, as for dating. I have been separated/divorced for more than 5 years. I am still not ready to date and also do not desire to do so. But every person is different. The one thing I can say is that it is NEVER a good idea to hurry into another relationship.
Maybe you can find a group or class that can help you deal with your pain and loss (A "Divorce Care" group at my church was extremely helpful to me). Find some books or resources that can help you deal with your pain and grief productively so you can grow stronger when you emerge from this dark place on the other side. So many people have similar stories and share how they have gotten through. Perhaps one of these books could help you as well, because we don't always have to go through hardships alone. Most of all, do not rush the process. Whatever time you need, you need. Allow yourself to have it (good days and bad) and at a certain point, you'll stand back and realize you are making GOOD progress and you'll know you're going to be OKAY.
Best of luck to you and please know that you are a caring, thoughtful mother and you should be very proud of that. You cannot make this man do right by your daughter no matter ho hard you try, so please don't put that burden on yourself. Perhaps he will, in the end, do right by her, but that is not your responsibility and the guilt and sadness that comes from his abandonment of your daughter is not yours to bear. That responsibility rests solely on HIS shoulders. As for BPD, (I have ADHD) everything is likely harder without treatment and/or meds, but know you still CAN do this if you make a plan, seek out help from others and explore resources to help you through this remarkably challenging times.
Let your commitment to being healthy for your daughter propel your decisions and reactions to this divorce and show her how much you care about her and that you will always do your part (and more) to keep her safe and ensure that she feels loved (as it is readily apparent you are committed to from your post). Keep up the great parenting and look for good moments (no matter how small) every day and celebrate them and maybe even keep them in a journal and revisit these successes from time to time.Best of luck to you and keep us posted!