1) better off

Let's work out an example. Suppose there is only one other good you may choose aside from gas, say bread. And your initial wealth is $20. Initial prices are $1 for each bread and $1 for gas. The budget lines is x + y = 20, where x is the amount of gas you consume and y is that of bread. (10, 10) is attainable since it's on the budget line.

Now suppose the price of gas rises to $2 and you are compensated $10 by the government. The budget line is now 2x + y = 30. So (10, 10) is still attainable.

Since the second line is steeper and both lines intersect at the point (10, 10) which is not the upper left corner of the feasible area, the second budget line gives better utility. (it's easier to see this on a graph)

2) cash is better (assuming your friend can buy other stuffs given that money).

Suppose tthe gift is worth $10. The price of another good A is $1. So the budget line is x + y = 10, where x is gift and y is good A. Given the gift, your friend is faced with the utility curve that touches the point (1, 0). Otherwise, he or she may choose any combination of consumption on the budget line which is usually better.

谢谢！强强