Get Sub-Category

I am trying to pull back a link and title of a post’s sub-category in a loop.

PARENTCAT1
-SUBCAT1
-SUBCAT2

PARENTCAT2
-SUBCAT4
-SUBCAT5

1) I want to return a link and title. i.e.
<a href="http://site/cat/subcat4">SUBCAT4</a>
2) I would like to change the SUBCAT4 name on the link
3) I should be able to specify the PARENTCAT2 or 1

This was kind of working, except I have to enter the category. I want it to use the category that the post is in and be able to only specify the PARENTCAT. A post will only be in ONE sub-category of the PARENTCAT

<?php

    // Get the ID of a given category
    $category_id = get_cat_ID( 'SUBCAT4' );

    // Get the URL of this category
    $category_link = get_category_link( $category_id );
?>

<!-- Print a link to this category -->
<a href="<?php echo $category_link; ?>" title="Category Name">SUBCAT4</a>

I tried this also but it just brings back the PARENTCAT2

<?php

    // Get the category name
    $category_name = get_cat_name(5); 

    // Get the ID of a given category
    $category_id = get_cat_ID( $category_name );

    // Get the URL of this category
    $category_link = get_category_link( $category_id );
?>

<!-- Print a link to this category -->
<a href="<?php echo $category_link; ?>" title="Category Name">SUBCAT4</a>