Ok, so to remove the tagline from the place it is now and put it under your blog title, do the following:
You should have class-header-header_main.php in your child theme folder
Find the area (near 175 line):
<?php ob_start(); ?>
<div class="<?php echo $logo_class ?> pull-left">
<?php
do_action( '__before_logo' );
printf('<%1$s><a class="site-title" href="%2$s" title="%3$s | %4$s">%3$s</a></%1$s>',
apply_filters( 'tc_site_title_tag', 'h1' ) ,
apply_filters( 'tc_logo_link_url', esc_url( home_url( '/' ) ) ) ,
apply_filters( 'tc_site_name_text ', __( esc_attr( get_bloginfo( 'name' ) ) ) ),
apply_filters( 'tc_tagline_text ', __( esc_attr( get_bloginfo( 'description' ) ) ) )
);
do_action( '__after_logo' )
?>
and add under it
<?php bloginfo('description'); ?>
This adds the tagline under the blog title.
Then find:
//body > header > navbar actions ordered by priority
add_action ( '__navbar' , array( $this , 'tc_social_in_header' ) , 10, 2 );
}
add_action ( '__navbar' , array( $this , 'tc_tagline_display' ) , 20, 1 );
}
and remove
add_action ( '__navbar' , array( $this , 'tc_tagline_display' ) , 20, 1 );
}
This removes the tagline from the place it is now.
Regarding the search box you can add <?php get_search_form(); ?>
in your header under
<?php
//the '__header' hook is used by (ordered by priorities) : TC_header_main::$instance->tc_logo_title_display(), TC_header_main::$instance->tc_tagline_display(), TC_header_main::$instance->tc_navbar_display()
do_action( '__header' );
?>
but you may need to polish the look of it.