Do below changes in your code (in red) or you can replace your code with my changed code also

Controller

public with sharing class ProductSearchController
    {
    // the soql without the order and limit
     private String soql {get;set;}
     // the collection of products to display
    public List<Product2> Products {get;set;}
    // the current sort direction. defaults to asc
    public String sortDir
    {
    get 
    {
        if (sortDir == null)
        { 
        sortDir = 'asc';
        }
    return sortDir; 
    }
    set;
    }
 
  // the current field to sort by. defaults to  name
    public String sortField
    {
    get 
    {
    if (sortField == null)
        {
        sortField = 'Name';
        }
         return sortField; 
      }
    set;
    }
 
  // format the soql for display on the visualforce page
  public String debugSoql {
    get { return soql + ' order by ' + sortField + ' ' + sortDir + ' limit 20'; }
    set;
  }
 
  // init the controller and display some sample data when the page loads
  public ProductSearchController() {
    soql = 'select name,productcode from Product2 where name != null';
    runQuery();
  }
 
  // toggles the sorting of query from asc<-->desc
  public void toggleSort()
  {
    // simply toggle the direction
    sortDir = sortDir.equals('asc') ? 'desc' : 'asc';
    // run the query again
    runQuery();
  }
 
  // runs the actual query
  public void runQuery()
  {
 
    try {
      Products = Database.query(soql + ' order by ' + sortField + ' ' + sortDir + ' limit 20');
    } catch (Exception e) {
      ApexPages.addMessage(new ApexPages.Message(ApexPages.Severity.ERROR, 'Ooops!'));
    }
 
  }
 
  // runs the search with parameters passed via Javascript
  public string name{get;set;}
  public string productcode{get;set;}
  public PageReference runSearch() {
      //String name = Apexpages.currentPage().getParameters().get('name');
    //String productcode = Apexpages.currentPage().getParameters().get('productcode');
    soql = 'select name,productcode from product2 where name != null';
    if (!name.equals(''))
      soql += ' and name LIKE \''+String.escapeSingleQuotes(name)+'%\'';
    if (!productcode.equals(''))
      soql += ' and productcode LIKE \''+String.escapeSingleQuotes(productcode)+'%\'';
 // run the query again
    runQuery();
 
    return null;
  }

 
}
 

 VF Page:

<apex:page id="pg" controller="ProductSearchController" sidebar="false">
  <apex:form id="frm">
  <apex:pageMessages id="errors" />
  <apex:pageBlock id="pb" title="Find Me A Product" mode="edit">
  <table id="tbl" width="100%" border="0">
  <tr> 
    <td id="tdSearch" width="200" valign="top">
 
      <apex:pageBlock title="Parameters" mode="edit" id="criteria">
          
          
      <apex:actionFunction id="afSearchServer" name="searchServer" action="{!runSearch}" rerender="results,debug,errors"/>
    <table cellpadding="2" cellspacing="2">
      <tr>
        <td style="font-weight:bold;">Name<br/>
        <apex:inputText id="name" value="{!name}" onkeyup="doSearch();"/>
        </td>
      </tr>
      <tr>
        <td style="font-weight:bold;">Product Code<br/>
        <apex:inputText id="productcode" value="{!productcode}" onkeyup="doSearch();"/>
        </td>
      </tr>
      </table>
 
      </apex:pageBlock>
      </td>
 
<td valign="top">
 
    <apex:pageBlock mode="edit" id="results">
 
        <apex:pageBlockTable value="{!Products}" var="Product">
 
            <apex:column >
                <apex:facet name="header">
                    <apex:commandLink value="Name" action="{!toggleSort}" rerender="results,debug">
                        <apex:param name="sortField" value="name" assignTo="{!sortField}"/>
                    </apex:commandLink>
                </apex:facet>
                <apex:outputField value="{!Product.name}"/>
            </apex:column>
 
            <apex:column >
                <apex:facet name="header">
                    <apex:commandLink value="productcode" action="{!toggleSort}" rerender="results,debug">
                        <apex:param name="sortField" value="productcode" assignTo="{!sortField}"/>
                    </apex:commandLink>
                </apex:facet>
                <apex:outputField value="{!Product.productcode}"/>
            </apex:column>
 
          </apex:pageBlockTable>
       </apex:pageBlock>
 
    </td>
  </tr>
  </table>
 
  <apex:pageBlock title="Debug - SOQL" id="debug">
      <apex:outputText value="{!debugSoql}" />
               
  </apex:pageBlock>   
 
  </apex:pageBlock>
 
  </apex:form>
<!--I HAVE REMOVED YOUR JAVA SCRIPT FROM THE TOP AND ADDED BELOW SCRIPT-->
      <script>
        function doSearch()
        {
            
            pg:frm:pb:criteria:afSearchServer:searchServer();
        }
      </script>
 
</apex:page>

I have tested and it works fine for me.