Trouble creating a child functions.php, also how to remove the text "Home"?

First question: How do I get the text “Home” at the top of the homepage to disappear? (but not necessarily every title on each page…just the homepage’s “Home”).

Could you please elaborate a bit? I’m not seeing any home button or link at the top.

If you mean to hide a link on your navigation bar and only on home the page, you can use this:

(if you’ve setup your nav location via Appearance –> Menus)

.home #menu-item-68 { /* Change #menu-item-## to the correct number of your item; it'll be different for every site  */
    display: none;
}

(if you’re using the default, without setting up anything via Appearance –> Menus)

.home .page-item-114 { /* Change .page-item-## to the correct number of your item; it'll be different for every site  */
    display:none;
}

Without seeing the site, I can’t tell which item-number it has. You’ll have to either provide us a link to your site or find it out on your own.

————–

Second question:
I am having trouble getting the header image height taller. You see, I know that I can change the padding from 10% to something else (like posted in other posts) and it does enlarge it which is nice…but then in crops it. So basically I want to go into the functions.php and change the “416” to “550”. But every time I try to make a child page for functions.php my website says:

So . . . something like in this screenshot? Instead of teh default cropping 1280×416, you wish to change it to 1280×550, using the functions.php in your child theme. Is that correct?

Try putting this into your functions.php, remove the <?php ?> tags if you already have something in place.

<?php

function hemingway_custom_header_setup() {
	$header_args = array( 'height' => 550 );
	add_theme_support( 'custom-header', $header_args );
}
add_action( 'after_setup_theme', 'hemingway_custom_header_setup' );

?>

You could do

.home .post-title,
.home .post-title a {
    display: none;
}

————

Only the new codes. If you actually copy the entire parent theme’s functions.php there, it would mess everything up.

WordPress’s child theme functions.php doesn’t replace the parent theme’s functions.php; it loads in addition to the parent theme’s functions.php. The order is like this: child theme’s functions.php loads then parent theme’s functions.php loads.

More info here: http://codex.wordpress.org/Child_Themes#Using_functions.php