Hi Sandy. I'm sorry about your dog dying; I lost my 18 year old cat last June and I still miss her.
Losing weight is not easy and Type 2 makes it harder. But it can be done, and you are young which is a great time to work on becoming healthy. While I certainly understand your desire to get a new puppy I need to be honest and tell you that losing weight fast is not the way to go. Number one it's not healthy and number two when people go on drastic diets and lose a lot of weight really quickly they tend to put it all back on. I don't really understand what you mean by a paralyzed tummy but if you have a medical issue then it's really important to talk with your doctor and have him help you to figure out what would be a healthy way for you to lose weight. For many of us, reducing our carbs really helps us lose weight. Do you know what foods are high in carbs? If not, get a book or look online (calorieking.com is a good site) and get in the habit of counting your carbs and working to lower your intake. This would be foods like potatoes, pasta, bread, sweets, cereal, etc. But to get a specific diet, perhaps your doctor can refer you to a dietician or cde. Also your doctor may recommend a medication that will help with your insulin resistance (not diet pills...those are a very BAD idea!).
Perhaps you can talk with your dad and work out a deal by which you can get the puppy for staying on the diet for a certain amount of time or losing x amount of weight not all the weight you need to lose, because you might be tempted to try and lose it too fast. My own personal opinion? If you show him you are motivated to lose weight to be healthier maybe you should just get the puppy because you are you!!
Finally since you mentioned being Depressed I hope you are in counseling to help you feel better. (Depression can also lead to poor eating habits!)