vik88ki...my heart goes out to you. I too have BPD, along with other things, and trust issues stemming from the past (both from abandonment/neglect in my childhood and the ex I was in my first long-term live-in relationship with was doing shady **** online all the time, including talking to other girls and that's not including all the weirdness and lies in real life) and even though I've been with my partner for 16 months now I still feel my hackles raise whenever I walk into the room and see her on the computer. I know in my case part of it is the past, even though I know logically she wouldn't do anything to hurt me. I know this isn't necessarily the same as your situation (you caught your partner talking to other girls) but I can relate to 'using it against' her. And in my mind every time she's on the computer she's talking to people that could possibly take her away from me. Even today the thought crossed my mind that maybe I should just leave her, as it would be easier than having to deal with this **** every time. Argh. But then I know that my feelings and behaviors and reactions will never change unless I stick around and do something different. So there's no point in running. Plus I am truly in love with her, and I believe she is the one I want to marry, settle down with, etc etc.
I can only suggest to take it one thing at a time. Have you thought about going to an intermediary? A couples counselor or something? To have someone help the two of you learn better communication skills? That might help you understand each other better and put things into a different context so that it no longer feels like an attack each time the two of you differ on something. It might even help you see eye to eye on more things.
As for the rest...I have big issues too with always needing to be right. Sometimes I say the dumbest stuff just because I don't want to concede anything to the other person. And then as soon as it leaves my mouth, I think, Why did I say that? Today is exactly one of those days in fact. As for the trust stuff (on both parts), it's hard to rebuild trust and also it's hard to say how long it will take until you have truly rebuilt it. There is no time frame, no magic number. And it's frustrating when people don't understand what living with mental illness is like. Do you have supports in your life who DO understand and support you? Is your partner open to learning more about your illness?
With the housework: would it help you to break it up into smaller jobs so that you're not overwhelmed with the whole of 'housekeeping'? Like, one day do the vacuuming, the next day do the sweeping/mopping, the next day clean the bathroom, the next day do the laundry? Just a suggestion.
Hopefully some or any of this is helpful to you. Hang in there...I believe that you will be able to work it out one way or another. At the risk of sounding like a cheesy self-help guru or something, every day is new. It's hard to believe that sometimes. But with every day that passes we have the chance to do something different in situations that usually baffle or disempower us.
Lastly...do you have access to treatment for your illness? I am assuming by BPD you mean Borderline...what about DBT? I've taken it and it works well in a lot of situations, including trust issues and communication issues.