Show category description using category id

$ca = $category->load($category_id);
$ca->getProductCollection()->addCategoryFilter($ca)->addAttributeToSelect('*');
echo $ca->getDescription();

How to remove parent category path from sub category in magento URL?

Step 1:

Edit "/app/code/core/Mage/Catalog/Model/Url.php"
Around line 805 comment the below lines:

//if (null === $parentPath) {
//    $parentPath = $this->getResource()->getCategoryParentPath($category);
//}
//elseif ($parentPath == '/') {
    $parentPath = '';
//}

Step 2:

Now re-index Catalog URL Rewrites in Index Management. That's it you are done.

How to add category and manufacturer dropdown to the magento default search?

Here is the code to add category and manufacturer / brand dropdown to the magento default search. Just replace your "template/catalogsearch/form.mini.phtml" with the below code.

<?php
$category = Mage::getModel('catalog/category');
if(is_object(Mage::registry('current_category'))){
    $current_category_path=Mage::registry('current_category')->getPathIds();
}else{
    $current_category_path = array();
}
$category->load(Mage::app()->getStore()->getRootCategoryId());
$children_string = $category->getChildren();
$children = explode(',',$children_string);
$extra_options='';
foreach($children as $c){
    $selected = (in_array($c, $current_category_path))?'SELECTED':'';
    $extra_options.= '<option $selected . value="' . $c . '" ' . '>' . htmlspecialchars($category->load($c)->getName()) . '</option>' . "\n";
}
?>

<form id="search_mini_form" action="<?php echo $this->helper('catalogSearch')->getResultUrl() ?>" method="get">
   <fieldset>
        <legend><?php echo $this->__('Search Site') ?></legend>
        <div class="mini-search">
            <input id="search" type="text" class="input-text" name="<?php echo $this->helper('catalogSearch')->getQueryParamName() ?>" value="<?php echo $this->helper('catalogSearch')->getEscapedQueryText() ?>" />
            <select name="cat">
            <option value="">Select Category</option>
            <?= $extra_options ?>
           </select>    
           <?php $product = Mage::getModel('catalog/product');
$attributes = Mage::getResourceModel('eav/entity_attribute_collection')
->setEntityTypeFilter($product->getResource()->getTypeId())
->addFieldToFilter('attribute_code', 'manufacturer');
$attribute = $attributes->getFirstItem()->setEntity($product->getResource());
$manufacturers = $attribute->getSource()->getAllOptions(false);?>
 <select name="manufacturer" id="manufacturer" class="" title="Manufacturer" >
          <?php foreach ($manufacturers as $manufacturer): ?>
            <option value="<?php echo $manufacturer['value'] ?>"><?php echo $manufacturer['label'] ?></option>
   <?php endforeach; ?>
       </select>
           <input type="submit" value="Go" style="border: 1px solid #808080;" alt="<?php echo $this->__('Search') ?>" />
            <div id="search_autocomplete" class="search-autocomplete"></div>
           <script type="text/javascript">
            //<![CDATA[
                var searchForm = new Varien.searchForm('search_mini_form', 'search', '<?php echo $this->__('Search Entire Site...') ?>');
                searchForm.initAutocomplete('<?php echo $this->helper('catalogSearch')->getSuggestUrl() ?>', 'search_autocomplete');
            //]]>
            </script>
        </div>
    </fieldset>
</form>  

How to get Parent category id in Magento?

Here is the code to get parent category id.
$current_category = Mage::registry('current_category');
echo $parent_id = $current_category->parent_id; 

Magento how to get category id in product page?

<?php $categories = $_product->getCategoryIds(); ?>
<?php foreach($categories as $k => $_category_id): ?>
    <?php $_category = Mage::getModel('catalog/category')->load($_category_id) ?>
    <a href="<?php echo $_category->getUrl() ?>"><?php echo $_category->getName() ?></a>
<?php endforeach; ?> 

This will load and link all of the categories that the product is found in.