How could I improve this vegetarian fat loss & muscle building diet?

Good job on the organic milk and eggs. You don't have to be vegetarian, though. Meat is fine. You can eat grass-fed, free-range beef, chicken, pork, and turkey. You don't have to worry about the fat in nuts. I personally don't believe the saturated fat theory, but even if you do, nuts have polyunsaturated fats and monounsaturated fats. Milk, if not raw, is a good source of protein, but is not really a good source of much else, as all the nutrients are killed off through pasteurization. I assume by whole grains, you mean whole grain bread. I would switch to Ezekiel bread, or another kind of sprouted grain bread if I were you, as sprouted grains are better for you than whole grains. You are doing a good job on the protein, organic foods, and supplements. If you are limiting your fat intake, which I assume by you saying you only eat low-fat nuts, you don't need to. Fat keeps you satiated, so if you skimp on the rice, but put a teaspoon or two more of coconut oil on your broccoli, you should have plenty of energy and feel full (although do what works best for you). Also, coconut oil is an amazing food. It decreases your appetite and improves thyroid function, which will speed up a slowed metabolism. Overall, good job, you get an A+... :D