This is basically what I am doing for my PhD (cell specialization), or more precisely how undifferentiated cells become neurons.
To the first question: Bob55’s answer has many elements of truth, but it is a little more general than that. HOX genes and integrins are only part of it, albeit in my field a rather large part of it.
In developmental biology, there is a concept of a morphogen – that is something (not necessarily a gene or gene product, but it can be) which has a concentration gradient across a field of cells. Examples of this may be simple things like calcium, retinoic acid, or light. More complicated things may be the ribonucleoproteins determining anteroposterior axes in the fruitfly embryo. HOX genes can certainly be included for some developmental processes. This morphogen will activate or repress different genes at different concentrations. This sets up distinct domains of gene expression, which relatively easy to subdivide as the tissue expands and starts to develop. Each subdivision triggers a new cascade of gene expression and eventual cellular specialization. Eventually, it can be subdivided to a cellular level.
Polarity is another issue. This is usually determined at very early stages. Usually, by the time you have a few cells, they know which way is up, down, left, right, front, and back. This is usually a stochastic event between a pair of cells. Each produce a receptor on its membrane and a ligand for the receptor on the neighboring cell. There is also feedback – the activated receptor means more receptor and less ligand is made. One cell, by chance, usually produces more receptor, and wins the repression battle. That one cell ends up producing all of receptor and the other cell all of the ligand.
In my field (sorry to get longwinded here, but it is late and I get excited to see questions like this), a group of cells in the embryo is first cordoned into an eye precursor by morphogens and subsequent gene cascades. This leads to the expression of a homeotic gene, eyeless and its partners, during embryonic development, which specifies the cells as retinal cells (they now will no longer form leg cells). This packet of cells goes on to form a flat disc of cells in the larva, where all of the interesting development starts to happen. Gradients are set up to determine dorsal from ventral and anterior from posterior (the cells know top from bottom already). Starting at the posterior of the disc, cells differentiate first into neuronal photoreceptors in a wavelike fashion. These recruit other cells in a very orderly fashion to be photoreceptors and accessory cells like pigment cells and lens-secreting cone cells. During pupal life, the cells start to adopt adult morphology to form the precisely structured 800 facet compound eye.
To answer the second question, until the process of gastrulation, which occurs at different times early in development depending on the organism, all of the cells of the embryo have totipotency, that is the potential to form any tissue (or another whole organism). So, yes it is possible to take one of those cells and make a twin. After gastrulation, most cells lose totipotency, but may still retain pluripotency, that is the ability to make many (but not all tissues). In mammalian embryos after gastrulation the inner cell mass of the gastrula are called embryonic stem cells (or ES cells). These cells have pluripotency to form any tissue in the embryo, but not the placenta or other associated structures.
These cells in mice have been successfully cultured for years now. You can (basically) grow them in a petri dish for as long as you need. While it is impractical to make each one into a separate mouse, it is pretty easy to introduce them into another gastrula, thus forming a chimeric mouse, which can be bred out using genetic strategies.
Could human ES cells be cultured? Absolutely, and they already are. It is not my field, but it is a big political argument about how many different lines, their exact pluripotency, their exact culture conditions, and a host of other things. You may remember Bush put some strict limitations on federal funding for research on stem cells, and that is proving to be a rather large hinderance on this kind of research.