Another thing I struggle with is communication with him. I am the type of person who wants to solve an issue immediately as it comes up. If he is in a mood and treats me disrespectfully, I find it hard to just say nothing and let the rage pass before we discuss it, especially when the rage lasts a long time. Does anyone have advice for this as well?
He really is trying to be better and I love him for it. It just hurts when he says hurtful things and isn't acting like himself. Whenever he is in a mood he can never recognize it himself and says "this is how I always am". I get frustrated because I want to help him get out of the mood but he won't acknowledge he is in one. I know pwBPD have difficulty recognizing their emotions and it is not his fault, but I just feel frustrated and desperate. 
Sorry this was rambling, we had a bad day yesterday after a long stretch of good days and I was feeling a bit discouraged. I don't want the house I grew up in to be the home I create for myself now. I just don't like feeling like I am walking on eggshells and having to orchestrate things so he doesn't get upset.  

It's good that you see the connection between the family you grew up in and the relationship you are in now. It's not possible to change your H's behavior but on your part, you can work on your emotional reaction to his behavior. 
I grew up with a mother with BPD and I also have had therapy to help me deal with this, but I also recognize when I've reacted to someone from that perspective when it isn't the same situation. 
If you are open to looking at ACA 12 steps- this has helped me a lot- working with a sponsor on my own reactions and behavior. The family dynamics with a disordered parent are similar to that of a family with a parent with alcoholism. In fact, some people call ACA- adult children of alcoholics and dysfunction. 
One aspect to work on is boundaries- what are our feelings and what are the other person's feelings. Walking on eggshells, feeling the need to "fix" things right away- these were survival behaviors for children growing up in a dysfunctional home. They may have been "functional" in that family setting but out in the adult world, they are not. ACA helped me to look at these behaviors and make some changes on my part. 
I don't think the feelings go away entirely, all the time but they do get better and we get better at recognizing them. 
Your goal is to be able to recognize your own reaction to your H's bad moods and manage that feeling. If your H is in a "mood" - those are his feelings. If you feel the urge to help him manage his moods- that is our of your own fear of them. You can learn to manage that better while also letting him learn to manage his own moods. 
Talking during these episodes may not help. They are how someone with BPD manages their uncomfortable feelings. However, you don't have to tolerate being the target or having him say mean things to you. One option is to tell him you love him and know he can manage this and that you will be available later. Let him have his own bad mood. Think of things you can do for yourself when he's in a bad mood- go for a walk, listen to music, call a friend and chat.