Answer:
12. Needn't
13. Can't
14. May
15. Should
Step-by-step explanation:
12. The speaker is saying how they can handle it by themselves; they don't need another person to tag along (the person physcally could, but they don't have to).
13. The girl who broke her leg is physically unable to go, whether people want her to or not.
14. The speaker is asking permission to do something, even though they are aware that they physically could at anytime.
15. The speaker is telling someone that it would be good 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)