An over-simplified model of cell division and its relation to stem cells
You have a few misconceptions about stem cells, I will try to explain where they are. 
First of all, cells are not independent. They influence each other with signals and secreted messenger substances. If you look at a human embryo the state of totipotency (where all cells can differentiate into each cell type of the body) ends after 3-4 days when the blastocyte has developed. This structure already has a differentiation into a inner cell mass (embryoblast) and a outer cell layer (tropoblast).
If all cells would simply divide on their own, this would rather look like bacteria dividing in culture media, but not as a forming, multicellular organism.

From these few cells in the beginning all cell types of the body have to develop (around 200, if I remember the number correctly). To reach this, the pluripotent stem cells need to proliferate and differentiate (develop into more specialized cell types) quite a lot. So embryonic stem cells do proliferate. Adult stem cells are not proliferating at the same rate a embryonic, but they still do. A good example are the melanocyte stem cells (which are already pretty specialized as they can only further differentiate into melanocytes) which replenish the melanocyte population of the hair bulb at the beginning of each hair cycle. Differentiated cells can divide as well, when necessary, think only about wound healing.

Question 2: Stem cells (especially adult stem cells) need very defined living conditions. This includes specific signalling molecules (Oct or nanog are examples for that) to stay stem cells. If this signalling is missing (and it does in the normal human body) then these cells either differentiate or (more likely) die.
The only stem cell transplant which takes place regulary is the transfer of hematopoietic stem cells to leukemia patients. There the problem is to keep the immune system from atacking these cells as foreign.