Cheese/Mad

Cheese = Poison?
Cheese is not ordinarily thought of as a deadly weapon.

Also, it is true that "poison" could be drawn in a picture in the form of a bottle with a skull and crossbones on it.

But given that a very common method of killing rats and similar pests is the use of poisoned bait, and cheese is a popular form of bait for rodents, the wedge of cheese in Bang's word balloon listing methods of killing could be taken as standing for poison as a means of killing.

Starving Hoboes?
While the word "tramp" has a sexual connotation, it also has an original meaning, of a poor person who wanders from one town to another. It seems not unreasonable that if a little girl heard her mother using that word in the former sense, she might take it as having the second meaning. Since poor people sometimes do not even have enough to eat, it is neither surprising nor surreal that a young girl might offer to share the food she happened to have on her with someone she has heard is an unfortunate.