1. What assumption about the production technology allows us to conclude that the demand curve for labor is negatively sloped?

Recall the definition of the demand curve for labor as the quantity of labor that should be hired to equate the marginal product of labor to the real wage, w/p. The demand curve slopes down because the marginal product of labor declines as the quantity of labor employed rises, with capital fixed. This property of the production function is known as the law of diminishing marginal productivity, or the law of variable proportions.
Yet another way to answer the question is to note that the level of the demand curve -- the marginal product of labor -- is the slope of the production function when plotted againts labor. So, if we assume that the slope of the production function declines as employment rises, the demand curve for labor will has a negative slope. One way of saying this is that the production function is assumed to be concave.