There are certainly some girls who have unusually early breast development (premature thelarche) or premature development of underarm and pubic hair (premature adrenarche). Most girls who have either of these are healthy, and they don t continue on through the other stages of puberty until they reach the appropriate age (most girls start their periods within a year of when their mother started). There is a medical problem called 'precocious puberty', however, and girls who have this will have early breast and pubic hair development, and continue on very rapidly to having periods. They also will have the growth spurt that is typically associated with puberty. Since you have described all three of these changes, I think it is important to have your daughter evaluated to make sure she does not have precocious puberty.
There are a number of causes of precocious puberty, most of which are quite treatable. It would be helpful, particularly for psychological reasons, to prevent a girl from starting her periods at seven or eight years of age. Also, as you suggested, growth in height slows down significantly once a girl starts her period. A girl who started her period at seven or eight would likely end up extremely short.
If your daughter is overweight, it may be that she does not truly have breast development, but just excess fat tissue. Her doctor can help to sort this out by examining her. If there is any concern about early pubertal development, there are blood and xray tests that can help to clarify whether her development is abnormal.