Part 1: I HAVE A LANGUAGE CHALLENGE FOR YOU. It seems that is a Marjorie Kinnan Rawlings and that the president, Dr. Anna Lillios, has published her email address online on this page: http://rawlingssociety.org/contact Why not email her and ask? You have nothing to lose. I think there is a good chance you will get an answer. Write in your own English as best you can. It doesn't have to be perfect. Don't be shy about it. My GUESS is that she will like hearing that Chinese students are studying "The Yearling" and will answer. Say something like this: "Dr Dr. Lillios I am a Chinese student studying English and we are reading 'The Yearling.' It has many words I cannot find in the dictionary. Could you help me one of them? " ... and continue with your question just as you have asked it here--just the part about "bay-head."

The phrase 'pushing through the bay-heads' gives the impression that the bay-heads are a type of vegetation that he is imagining his mother making her way through. 'Bay' is a type of bush, so maybe it means the heads of the bushes? Bay seems to grow in the area, according to Dan's quote from elsewhere in the book.

Part 3: "In spite of himself" means he did not want to laugh at his mother, he was trying not to laugh, but despite trying his best, the idea was so funny he started laughing anyway. "In spite of one's self" means trying but failing to keep control of one's emotions. Often, the emotion is laughter. In "A Visit from Saint Nicholas," the speaker finds Santa Claus so funny that "I laughed when I saw him in spite of myself." In one of the Wizard of Oz books: "The croaks that came at the end of the song were so terrifying that the Scarecrow shivered in spite of himself." He tried not to shiver but he shivered anyway. "In spite of himself Clarke felt inquisitive"--curiosity got the better of him.

Part 2: You didn't say, but should have said, that you are reading "The Yearling," a 1938 bestseller by Marjorie Kinnan Rawlings, that is set in the "Big Scrub" area of 1870s Florida. Part of its interest to U.S. readers is that it is about a time and place in the U.S. that most of us know little about. It is FULL of vocabulary, phrases, and colloquialisms that most native English speakers do not know. I don't know what a bay-head is, I have to guess from context. This is probably what most native English-speakers need to do. The "head of a bay" is the most inland part. The setting of "The Yearling" is near Cross Creek and Silver Bay Springs, names I didn't know until I found http://www.npr.org/2011/07/21/138561573/on-location-the-central-florida-of-the-yearling . Now I come from the New England area and think of a "bay" in terms of water, but it seems perfectly clear that in this context, it means the densely overgrown land NEAR the head of the bay--territory that would require constant work pushing through close-grown shrubs. Fortunately I have a searchable PDF of "The Yearling" so I can check for other places where the word is used and in another place I find: "The deer were feeding on the tender growth, bud of sweet bay and of myrtle, sprigs of wire-grass, tips of arrowroot in the ponds and prairies, and succulent lily stems and pads. The type of food kept them in the low, wet places, the swamps, the prairies and the bay-heads." So a bay-head is a kind of low, wet land, that grows the kinds of shrubs deer like to eat.