Add custom text field to field settings form In the admin when click edit on Manage Fields page it shows settings for that field. Can we add a custom property (checkbox) here like the required checkbox?

For example, for a decimal field it show 'Label' textbox, 'Required field' check box, 'Help text' textarea and so on. Can I add my own checkbox or textfield to this form?
What you'd have to do is define your own field type in a module, then use hook_field_settings_form to set the global options for that field (or hook_instance_settings_form for a specific content type's implementation). 

Here's a general guide to creating your own field in Drupal 7, but to get a handle specifics of creating a custom settings option take a look at other modules' implementations of it.

For example, Drupal Commerce makes a set of options from existing customer profiles:

function commerce_customer_field_settings_form($field, $instance, $has_data) {
$settings = $field['settings'];
$form = array();

if ($field['type'] == 'commerce_customer_profile_reference') {
  $options = array();

// Build an options array of the customer profile types.
foreach (commerce_customer_profile_type_get_name() as $type =&gt; $name) {
  $options[$type] = check_plain($name);
}

$form['profile_type'] = array(
  '#type' =&gt; 'radios', 
  '#title' =&gt; t('Customer profile type that can be referenced'), 
  '#options' =&gt; $options, 
  '#default_value' =&gt; !empty($settings['profile_type']) ? $settings['profile_type'] : 'billing', 
  '#disabled' =&gt; $has_data,
 );
}

return $form;
}


Yours sounds more simple - but the same general idea. Get the field and instance data, do something with it, make another mini form and return it so it gets rendered.