echo selected value from dropdown

This is purely an HTML issue. The problem is that you’ve tried to indicate the selected item by setting a value="" attribute on the <select> element. That’s not how <select> works.

To mark an option as selected by default the <option> tag needs to be given the selected attribute.

For example, you are outputting this:

<select id="j_applicationCategory" name="j_applicationCategory" value="UtilitiesApplication" style="width: 100%" />
    <option value="GameApplication">GameApplication</option>
    <option value="UtilitiesApplication">UtilitiesApplication</option>
    <option value="DesignApplication">DesignApplication</option>
    <option value="ReferenceApplication">ReferenceApplication</option>
</select>

You need to output this:

<select id="j_applicationCategory" name="j_applicationCategory" style="width: 100%">
    <option value="GameApplication">GameApplication</option>
    <option value="UtilitiesApplication" selected>UtilitiesApplication</option>
    <option value="DesignApplication">DesignApplication</option>
    <option value="ReferenceApplication">ReferenceApplication</option>
</select>

Also note that I have removed an inappropriate closing / from the select tag.

WordPress has a utility function, selected(), that makes outputting this attribute based on the current value easier:

echo '<select id="j_applicationCategory" name="j_applicationCategory" style="width: 100%">';
echo '<option value="GameApplication" ' . selected( $j_applicationCategory, 'GameApplication', false ) . '>GameApplication</option>';
echo '<option value="UtilitiesApplication" ' . selected( $j_applicationCategory, 'UtilitiesApplication', false ) . '>UtilitiesApplication</option>';
echo '<option value="DesignApplication" ' . selected( $j_applicationCategory, 'DesignApplication', false ) . '>DesignApplication</option>';
echo '<option value="ReferenceApplication" ' . selected( $j_applicationCategory, 'ReferenceApplication', false ) . '>ReferenceApplication</option>';
echo '</select>';