how to reference a YAML "setting" from elsewhere in the same YAML file?

Question

I don't think it is possible. You can reuse "node" but not part of it.

bill-to: &id001
    given  : Chris
    family : Dumars
ship-to: *id001

This is perfectly valid YAML and fields given and family are reused in ship-to block. You can reuse a scalar node the same way but there's no way you can change what's inside and add that last part of a path to it from inside YAML.

If repetition bother you that much I suggest to make your application aware of root property and add it to every path that looks relative not absolute.

Yes, using custom tags. Example in Python, making the !join tag join strings in an array:

import yaml

## define custom tag handler
def join(loader, node):
    seq = loader.construct_sequence(node)
    return ''.join([str(i) for i in seq])

## register the tag handler
yaml.add_constructor('!join', join)

## using your sample data
yaml.load("""
paths:
    root: &BASE /path/to/root/
    patha: !join [*BASE, a]
    pathb: !join [*BASE, b]
    pathc: !join [*BASE, c]
""")

Which results in:

{
    'paths': {
        'patha': '/path/to/root/a',
        'pathb': '/path/to/root/b',
        'pathc': '/path/to/root/c',
        'root': '/path/to/root/'
     }
}

The array of arguments to !join can have any number of elements of any data type, as long as they can be converted to string, so !join [*a, "/", *b, "/", *c] does what you would expect.

Another way to look at this is to simply use another field.

paths:
  root_path: &root
     val: /path/to/root/
  patha: &a
    root_path: *root
    rel_path: a
  pathb: &b
    root_path: *root
    rel_path: b
  pathc: &c
    root_path: *root
    rel_path: c

I've create a library, available on Packagist, that performs this function: https://packagist.org/packages/grasmash/yaml-expander

Example YAML file:

type: book
book:
  title: Dune
  author: Frank Herbert
  copyright: ${book.author} 1965
  protaganist: ${characters.0.name}
  media:
    - hardcover
characters:
  - name: Paul Atreides
    occupation: Kwisatz Haderach
    aliases:
      - Usul
      - Muad'Dib
      - The Preacher
  - name: Duncan Idaho
    occupation: Swordmaster
summary: ${book.title} by ${book.author}
product-name: ${${type}.title}

Example logic:

// Parse a yaml string directly, expanding internal property references.
$yaml_string = file_get_contents("dune.yml");
$expanded = \Grasmash\YamlExpander\Expander::parse($yaml_string);
print_r($expanded);

Resultant array:

array (
  'type' => 'book',
  'book' => 
  array (
    'title' => 'Dune',
    'author' => 'Frank Herbert',
    'copyright' => 'Frank Herbert 1965',
    'protaganist' => 'Paul Atreides',
    'media' => 
    array (
      0 => 'hardcover',
    ),
  ),
  'characters' => 
  array (
    0 => 
    array (
      'name' => 'Paul Atreides',
      'occupation' => 'Kwisatz Haderach',
      'aliases' => 
      array (
        0 => 'Usul',
        1 => 'Muad\'Dib',
        2 => 'The Preacher',
      ),
    ),
    1 => 
    array (
      'name' => 'Duncan Idaho',
      'occupation' => 'Swordmaster',
    ),
  ),
  'summary' => 'Dune by Frank Herbert',
);