On one hand, I'm a big fan of not rushing into things when you're very young, but on the other, you do sound mature and thoughtful, and I wish you the best in this relationship.
Now to the nitty-gritty: You have two stressors compounding an already stressful situation - your parents' demand to move back home along with your communication problems with the fiancé all on top of wedding preparations! Oy! I don't envy you.
That said, here's my take on each of the two situations.
About Your Parents...
The parental interference is somewhat easier to tackle. For the most part, I think they're being unreasonable and manipulative: at best trying to get one last stint of their little girl at home before she leaves to set up her own household, or at worst attempting to assuage personal guilt or undo some supposed lack on their part in teaching you a value system that frowns on "living in sin" before marriage.
If they can afford to pay for the wedding and previously offered to do so, reneging on that offer is a gross breach of trust that is hurtful to you, and communicating that to them may be illuminating.
However, there is a little voice inside me that, in trying to see their side of it, gives the benefit of the doubt and says that you should be as accommodating as possible to their requests, since they ARE paying for the wedding. Still (yet another qualification! And I said this was the SIMPLE problem...), moving back home would probably be highly impractical for you after living with your intended for so long, and if you point this out to them you might be able to get them to see from a perspective they hadn't considered.
About Communication With Your Fiance...
As to communication between you and your fiancé, you yourself have distilled much of the information you need to improve the situation - perhaps you just need a little prodding to see how to apply this information.
Take Care of Little Things AS They Come Up
One way to cut down on occurrences of small arguments turning into huge fights, often exacerbated by interjection of past indiscretions or failings, is to bring up little things as soon as they come up. In the short run, this could bring a flood of nitpicky comments, but if you both consciously make an effort to do this, you could find easy ways to alter some habits, cease others, and simply learn to live with still more - in either case, once they've been brought up and dealt with them, they are no longer fair game in other situations.
Again, this takes some getting used to and you both have to understand and agree to do it, but it makes living with one another much easier in the long run.
Avoid Calling Each Other Names
Another important point is to try and avoid calling each other names or using blanket statements about character flaws (you noted as an example your fiancé calling you a hypocrite). Unless one is the pure embodiment of a certain trait, and no human on this earth is, this is NEVER fair and just as rarely true.
In this case, try not to present this in a one-sided attack that will put him on the defensive. If you can't think of any instances where you have done the same thing, pretend that you don't recall any specific infractions of his either, but just say that you know you've both done it, you're both guilty, and you should both try to stop. As with most things, this will take some effort and a change will not be accomplished overnight, but the benefits of changing your patterns of speech to each other are priceless.
Think About What You Are Thinking
One of the things I learned in a past relationship was applying cognitive behavioral methods to fixing a relationship from the inside. That is, literally thinking about what I thought about, trying to see the thought process through to its conclusion, taking the other person's perspective and thinking about how s/he would see my behavior, and eventually working on altering any destructive or negative behavior.
This is a personal endeavor, but it does not mean you must undertake it alone. Talking about the process with your partner, writing about it in a journal, sharing it with a friend - all of these are valid accompaniments to your healing and change.
Couple Counseling Could Help
To your credit, while admitting to some of your own and your fiancé's flaws, you note your mutual desire to communicate more effectively and make things better. While I hope the advice you receive from me and my colleagues is helpful, you should not be afraid to turn to professional help before or after you get married - guided counseling sessions with your partner can often be tremendously enlightening, and sometimes just a few meetings with a therapist can foster a lifetime of normalcy (I purposely don't say "tranquility" because life isn't always tranquil).
May you find all the happiness you seek and deserve!