Loads of posts I didn't read, but I hope you read mine.
First off, nothing HAS to turn into a fight if you sit down and calmly explain a plan of action. A division of chores around the house for example. It's tough to keep a cool head when you love someone, but if you approach it in a way where she doesn't have to feel like you're attacking her, and therefor have to go on the defense, then things and open communication would be less stressful.
It's an art not to push the other person into defensive mode, whenever I approached a group of women, and say something controversial (however true it may be), ALL of them will gang up on me in defense even if they have to throw all logic out of the window. They fight emotionally and not long before then it's an insult-fest. One I usually win.
So, plan of action?
1) Prepare a plan. List the things that currently bothers you and come up with a plan to solve them. Dividing up the chores that will give her enough "space" to do whatever she wants, study and still contribute to the house-hold. Don't talk about it, come up with a schedule both of you can stick to. Something like "The super nanny" would do with the kids to help around the house. A board of some kind where every task is listed on a day to have it completed. And since both of you aren't kids (well, she might still be in most ways), you could "reward" each other with sexual favors or back/foot rubs etc. From what you've said, she does sometimes do things around the house, but it really does sound like there's no structure as to who does what and when. An example would be to schedule who makes dinner. Mo/We/Fri would be your turn, Tue/Thu/Sat her turn, and Sunday you can maybe go out for lunch/brunch at a cheap enough place that doesn't sell lobster.
2) Not being able to go to fancy dinner dates or fun activities due to the money issue doesn't mean you can go out at least once a week to go to the movies or the theater. Both of which is relatively cheap enough to not break the bank and easily budgeted for. My one friend and his wife (who dated 9 years before tying the knot) has a WEEKLY date night, even with their little bundle of joy, to go out to the movies and maybe a bite to eat. Spur isn't THAT expensive dude.
3) Communication. Romance & Intimacy takes work. So does love. It's never easy, nor should it be. But I've seen that this generation is so quick to quibble over money/sex and break up without actually working on it, it's insane. This is a 2 way street and your attempts at communicating with her effectively without putting her on the defense and eventually leading to a fight will take some work from your side.
4) She's spoiled. There has been no real pressure put on her to actually grow up and out of the phase where everything is done for her or handed to her on a silver platter. You can't change that overnight nor can you pressure her into anything. It will be a gradual process. A process BOTH of you are willing to work on and perfect as time goes by. Oh... and do me a favor. Don't call her spoiled or a brat.
5) Keep calm. If she starts a fight (or wants to start a fight), just keep calm and think thrice before you say anything. Remember, you're the man in the relationship, you should dictate a plan of action and be open to discuss any changes she might feel this "plan" needs. If she's too immature to accept anything you come up with, ask her to help solve the problem and remind her that you love her with a passion and that you'd like to have a life with her.
IMO you didn't move too fast, nor did you make a mistake by moving in together. It's a part and transition for both of you to get to know each other more intimately. If, after 4 months, you want to throw in the towel and dismiss the last 4 years of your relationship, by all means, go ahead and do it. You're still young and you can do it if you wanted to, but please note what I said about it taking hard work.
It's a two-way street at the end of the day. If she's not willing, then you'd need to take action and end things rather. She's still young as well, and her age may be the thing that stands in the way. Women often don't want to feel "tied down" at that age and want to experience life etc. But speaking from personal experience (dating and family), if she's like this when she's studying, she'll be even worse when she's working and contributing financially. Better try to change it now while you can so that both of you grow together instead of apart.