it doesn't look like your missing anything, I've checke the docs but they are a little spartan to say the least...
If yuo wanted to simply display the field within a {exp:channel:form} SAEF it seems you can use the {field:var_name}
syntax, you should also be abel to use the variable tag pair syntax within the Cahnnel Form. However it sounds like you just want to use the options in a custom dropdown as you mention 'a filter'
You still have some options, but if you're dead set on extracting the optoins from the P&T Dropdown on a Channel Field (rather than recreating them in a snippet/global variable or summin) you could roll your own PHP Sql query to get them or attempt to use the EE Template API to get the options/render the field again using PHP.
Let me know if you want a concise example, I'll have to knock one together. However in the meantime these docs might get you on the right path...
Edit: Been a while since I dipped into the Channel Fields API, so I tried an example. The following snippet will get you the options (as a key => value array) for a P&T Dropdown, provided you know the Channel (1) and Field ID (3) :
<?php
//instansiate the channel entries API
ee()->load->library('api');
ee()->api->instantiate('channel_fields');
//assuming Channle has ID of 1
$channelSettings = ee()->api_channel_fields->setup_entry_settings('1', array());
//assuming dropdown is field ID of 3
$fieldOptions = empty($channelSettings['field_id_3']['options']) : array() ? $channelSettings['field_id_3']['options'];
?>