Hi lanaharvey,
From what you've said, I'm wondering: do you feel overwhelmed? Recovering from an abusive childhood takes time and work, and it can feel overwhelming at times.
--Despite keeping a very organized work environment, I am incapable of keeping my apartment organized to the point where it's almost "Hoarder's" status. Attempting to clean makes me feel uneasy and I cannot keep the place clean after cleaning for very long at all.
Sometimes it's a lot easier to take on a task like cleaning up a large area if you break it down into small tasks. What's one thing, even if it's a small thing, you could do today? Is there a bookshelf you could organize, or could you set aside 5 articles of clothing to donate?
--When people talk about their families I have to suppress an extremely jealous spiteful feeling
--I never fully emotionally invest in a (romantic or friendly) relationship in fear of betrayal or abandonment, keeping everybody "at arm's length"
--Although I have many friends I sometimes have a continued irrational "alone", or "empty" feeling
It's very normal to want what others have, especially when it comes to loving and supportive families. If you grew up in a house full of conflict (as many of us here have), it can be very hard to feel comfortable opening up to others. This is something worth exploring in therapy, as that can be a huge help.
--I get an indescribable "brain fog" occasionally where I am detached from the present in a weird way
When you find yourself detatched, what do you find yourself thinking about? Is there something that triggers this FOG?
You're a survivor, that's for sure! It takes a lot of courage to pick up and set out on your own at 18. You sound like a very strong person.
Am I harboring resentment for having to handle my own life completely on my own? Am I valid in having no "family" feelings toward them?
You could be harboring resentment for many reasons. I could see why you'd be angry that you felt like you had to leave. Many people are still very attached to their parents (financially and emotionally) at 18.
My take on feelings is that they're not wrong; they're your feelings, and it's healthy and natural to have them. What's more important is how you act out on those feelings. It's hard to feel an attachment to people that you don't feel emotionally connected with, even if they are your biological family.
How do you deal with answering questions from others like "What are you doing for Father's Day?" without completely bringing down the conversation or seeming imbalanced?
Most people only ask to make conversation. I'm honest with people, but don't get into a lot of detail. "We'll have a quiet day, as my father is in [state]." Then I'll try to turn the conversation back to the other person... . most people love talking about themselves.

I want to accept other people who have had easier lives rather then be prejudiced against those born wealthy or supported at home. I almost feel better than my peers for having to overcome more than they did. That's not fair or right. How do I shake these feelings?
This is where empathy comes in, and I think this is something else you might want to explore in therapy (or T as we call it). Remember that we're all responsible for our actions, but some people are impacted by circumstances beyond their control, like the families they were born into or illness.
I know therapy is definitely needed but do you guys have any advice from personal experience? Have any of you dealt with these issues after being raised by a BPD parent?
I'm a walking ad for T. It has helped me really see myself in a new light and recognize that while I couldn't have avoided some things in the past because of my upbringing, I can change how I behave going forward. It has helped me process the abuse (which I didn't see as abuse, BTW), and look at life from a healthier angle.
T is a lot of work. I've left my T's office feeling great at times, and I've left feeling sad and drained at other times. It's worth the effort, though. I'm a stronger and happier person than I was a few years ago.