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Problem : Why can a worker's budget constraint for leisure and all other goods only pivot around one end?

Recall that a budget constraint is a line connecting the two maximum points of consumption for each good. The maximum amount of leisure that a worker can consume is 24 hours in one day. This amount cannot increase, as there are only 24 hours a day. The worker's consumption of all other goods (AOG) can change, however, depending on the level of wages the worker is earning.

Problem : How does a decrease in wages affect workers' decisions between leisure and labor?

Unless we are at a very high wage level (in which case a backwards-bending supply curve could change things), a drop in wage will have three affects: a) a decrease in consumption of both leisure and AOG (income effect); b) a decrease in the consumption of AOG (substitution effect); c) and an increase in the consumption of leisure, since leisure becomes relatively less expensive when there are lower wages lost in its pursuit (also the substitution effect). This means that workers will work less and play more.

Problem : What set of circumstances would generate a backward-bending labor supply curve?

If the substitution effect outweighs the income effect for low wage levels, and the income effect outweighs the substitution effect for high wage levels, there will be a backward-bending labor supply curve.

Problem : Why is there a tradeoff between leisure and consumption, and not between labor and leisure?

Leisure and consumption are both normal goods, while, for most people, labor is a means to an end rather than a good in and of itself. (Most people, given a choice between labor and leisure, would choose leisure, which wouldn't give us a very useful budget constraint to worth with). Thus, the tradeoff is between leisure and the things you can buy if you choose labor (consumption). If you choose leisure, you don't work as much, and so you can't buy as much. If you choose consumption, you have to work more, and you can't play as much.

Problem : What happens to labor supply if the wage increases, and the labor supply curve is backward-bending?

It is uncertain, since we don't know where we are on the curve. If we are at a point before the bend, then an increase in wages will cause an increase in labor supply. If we are at a point after the bend, then an increase in wages will cause a decrease in labor supply.
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