Assumptions about indifference curves and reasoning lies behind them

There are a few standard assumptions about what an indifference map can and cannot look like. Which are these assumptions, and what reasoning lies behind them?

Assumptions about indifference curves:

1) A consumer always prefers points that are to the northeast of a given point and, vice versa, prefers a given point to all points southwest of it. This depends on the assumption of non-satiation. This implies that an indifference curve cannot slope upwards.

2) For the same reason, indifference curves that lie northeast of a given indifference curve must correspond to a higher level of utility.

3) The slope of an indifference curve diminishes as quantity increases (i.e. to the right in a graph). This depends on the diminishing marginal utility of consumption.

4) Two indifference curves cannot intersect. This is easy to see if one thinks of indifference curves as elevation contours on a map.