Sorry I missed your initial posts, I've been away. The mods frown on thread bumps, but it may have paid off this time.
I don't know if I can make it clear, but I have an added field in my comment form, I can briefly explain what I've done. To add the field, hook the filter 'comment_form_default_fields'. Your filter function is passed an associative array of form field html. Simply add your field's html to the array and return the entire array. If you're not happy with the order of the fields, copy the values into a new array in the order you like and return the new array.
Next you need to get the form POST data into the DB. I used the 'wp_insert_comment' action, though other comment hooks probably could work as well. Get your field value from $_POST['your_field_name'] and save it somewhere. I used the comment meta table.
I can't tell you how to reflect your saved value in the comment output because my field was an operational setting which does not effect comment output. If you can't figure this part out, let me know and I'll see if I can suss out an approach.
For reflecting the setting in the admin panels, I added a column to the comment table. There is a page in the Codex for adding a column to the posts table. All tables are based on the same root class, so the technique translates to any table. First you register your column by hooking the 'manage_edit-comments_columns' filter and add your column name as a new key/value in the array your are passed. Then return the array.
Once registered, to display the actual value in the column, hook the 'manage_comments_custom_column' action. Your action function is passed the column name and comment ID. If the column name belongs to you, that is your cue to echo out the column table cell html content for the particular comment.
How or if you administer this value depends on what it's purpose is. Some sort of AJAX technique is well suited for this. AJAX in WP is a large tutorial in itself. I will warn you that the usual AJAX techniques need to be modified for WP. If you are not enqueueing your script and hooking an ajax action, you are doing it wrong.
I fear all this is even more confusing than the info you've already reviewed. If you get stuck somewhere and have a specific question, post a code snippet of what you have so far and I'll try to help, but I can't write up a full tutorial. This will at least get you pointed in the right direction and provides some key information. Good luck!