Answer:
12. Needn't
13. Can't
14. May
15. Should
Explanation:
12. The speaker is saying how they can handle it by themselves; they don't <em>need</em> another person to tag along (the person physcally could, but they don't <em>have</em> to).
13. The girl who broke her leg is physically <em>unable</em> to go, whether people want her to or not.
14. The speaker is asking <em>permission</em> to do something, even though they are aware that they physically could at anytime.
15. The speaker is telling someone that <em>it would be good</em> for them to eat less sweets. "Shall" would mean that the person will eat less sweets, no matter if they want to or not. "Had better" is usually used as a threat, and does not fit this context.
(I italicized words that show the meaning of the correct answers)
