Thanks for the fast reply! I've made the suggested changes but the page is still not fully functioning properly and I'm not quite sure why. It appears that my sObjectDesc variable is still not being updated upon changing my selectList value; however, I can see that the search results are being updated (even though the results are the same upon each refresh).

If it helps-my long term goal is to emulate the functionality of the Salesforce page where you add/remove members from a group such as in the page you reach after clicking Setup>Manage Users>Public Groups>Edit (Any Group)

Thanks for your help, below is my updated code:

VFPage:

<apex:page standardController="Janela_de_Acesso__c" extensions="JDABP" sidebar="false">
<apex:sectionHeader title="Janela de Acesso"/>
<apex:form >
<div align="center">
<apex:messages styleClass="errorMsg"/>
<apex:commandButton action="{!Save}" value="Save"/>
<apex:commandButton action="{!SaveAndNew}" value="Save & New"/>
<apex:commandButton action="{!Cancel}" value="Cancel"/>
</div>
<apex:pageBlock >
<apex:pageBlockSection title="Informações" columns="2">
<apex:inputField required="true" value="{!Janela_de_Acesso__c.Name}" />
<apex:inputField required="true" value="{!Janela_de_Acesso__c.Data_Inicial__c}"/>
<apex:inputField required="true" value="{!Janela_de_Acesso__c.Status__c}"/>
<apex:inputField required="true" value="{!Janela_de_Acesso__c.Data_Final__c}"/>
</apex:pageBlockSection>
 
<apex:actionFunction immediate="true" name="searchServer" action="{!runSearch}" reRender="Available2,Available3">
    <apex:param assignTo="{!searchText}" name="searchB" value="" />
    <apex:param assignTo="{!sObjectDesc}" name="pesq" value="" />
</apex:actionFunction>

<apex:pageBlockSection title="Membros da Janela de Acesso" columns="1" id="Available_BPs">

<apex:panelGrid columns="2" cellpadding="5px">
<apex:pageBlockSectionItem labelStyle="vertical-align: middle" >
<apex:outputLabel value="Pesquisa:" for="SearchList"/>
<apex:selectList title="Pesquisa" size="1" id="pesquisa" onchange="doSearch();">
<apex:selectOptions value="{!items}" id="pesquisa2"/>
<script type="text/javascript">
      function doSearch() {
        searchServer(
          <!--document.getElementById('{!$Component.SearchBox}').value,-->
          document.getElementById('{!$Component.pesquisa}').options[document.getElementById('{!$Component.pesquisa}').selectedIndex].text);
      }
</script>

</apex:selectList>
</apex:pageBlockSectionItem>
<apex:pageBlockSectionItem labelStyle="vertical-align: middle">
<apex:outputLabel value="for:" for="SearchBox"/>
<apex:inputText id="SearchBox" onkeyup="doSearch2();">
<script type="text/javascript">
      function doSearch2() {
        searchServer(
          document.getElementById('{!$Component.SearchBox}').value,
          <!--document.getElementById('{!$Component.pesquisa}').options[document.getElementById('{!$Component.pesquisa}').selectedIndex].value);-->
      }
</script>

</apex:inputText>
</apex:pageBlockSectionItem>
</apex:panelGrid>
<apex:panelGrid columns="2">
<apex:outputLabel value="Membros Disponíveis" style="font-weight:bold"/>
<apex:outputLabel value="Membros Selecionados" style="font-weight:bold"/>
</apex:panelGrid>
<apex:panelGrid columns="3">
<apex:outputPanel id="Available">
<apex:selectList title="Membros Disponíveis" size="14" id="Available2">
<apex:selectOptions value="{!searchResults}" id="Available3" />
</apex:selectList>
</apex:outputPanel>
<apex:selectList title="Membros Selecionados" size="14" id="Selected"/>
</apex:panelGrid>
</apex:pageBlockSection>
</apex:pageBlock>
</apex:form>

</apex:page>

 Controller:

public class JDABP {    
    
    // the soql without the order and limit
    private String soql {get;set;}
    // the collection of contacts to display
    public List<sObject> results {get;set;}
    public String searchObject;
    public Janela_de_Acesso__c j;
    public List<SelectOption> searchResult = new List<SelectOption>();
    public String searchText=Apexpages.currentPage().getParameters().get('searchB');
    public String sObjectDesc=Apexpages.currentPage().getParameters().get('pesq');

    
    //Constructor to apply fields
    public JDABP(ApexPages.StandardController controller) {
        j = (Janela_de_Acesso__c) controller.getRecord();
        j.Name='|';
        j.Status__c='Ativo';
        j.Data_Inicial__c=System.now();
        j.Data_Final__c=System.today()+30;        
        //j.RecordTypeId = Label.Janela_de_Acesso_BP_RecordType;
        searchText = '';
        sObjectDesc = 'Business Plans';
        searchObject = 'Business_Plan__c';
        soql = 'select Id, Name from ' + searchObject + ' limit 100';
        runQuery();

    }
    
    public List<SelectOption> getItems() {
        List<SelectOption> options = new List<SelectOption>();
            options.add(new SelectOption('Business Plans','Business Plans'));
            options.add(new SelectOption('Perfis','Perfis'));
            options.add(new SelectOption('Usuários','Usuários'));

        return options;
    }     
    
    public List<SelectOption> getSearchResults() {
        return searchResult;
    }
    
    // runs the actual query
  public void runQuery() {    
    try {
      results = Database.query(soql);
      searchResult.clear(); 
      for (sObject r : results){
          If (sObjectDesc.contains('Business Plans')){
              Business_Plan__c BPr = (Business_Plan__c) r;
              searchResult.add(new SelectOption(BPr.name,BPr.name));
          }
          Else If (sObjectDesc.contains('Perfis')){
              Profile Pr = (Profile) r;
              searchResult.add(new SelectOption(Pr.Name,Pr.Name));
          }
          Else If (sObjectDesc.contains('Usuários')){
              User Ur = (User) r;
              searchResult.add(new SelectOption(Ur.Name,Ur.Name));
          }
          Else {
              searchResult.add(new SelectOption(sObjectDesc,sObjectDesc));
          }         
      }
    } catch (Exception e) {
      ApexPages.addMessage(new ApexPages.Message(ApexPages.Severity.ERROR, soql+' '+e.getMessage()));
    }
 
  }
 
/*Method for troubleshooting
  public PageReference runReset() {
      searchResult.clear();
      searchResult.add(new SelectOption(sObjectDesc,sObjectDesc));
      return null;
  } 
*/ 
  // runs the search with parameters passed via Javascript
  public PageReference runSearch() {
 
    If (sObjectDesc.contains('Business Plans')){
    searchObject='Business_Plan__c';
    }
    Else If (sObjectDesc.contains('Perfis')){
    searchObject='Profile';
    }
    Else If (sObjectDesc.contains('Usuários')){
    searchObject='User';
    }
 
    soql = 'select Id, Name from ' + searchObject + ' limit 100';
    if (!searchText.equals(''))
      soql += ' and Name LIKE \''+String.escapeSingleQuotes(searchText)+'%\'';
    
 
    // run the query again
    runQuery();
 
    return null;
  }

        
    //Override Save & New Functionality
    public PageReference SaveAndNew() {
        try{
            insert j;
        } catch (DMLException e) {return null;}
        PageReference pr = new PageReference('/apex/JDABP');
        pr.setRedirect(true);
        return pr;
    }

}