<apex:page Controller="actionSupportController">
   <apex:form >
       <apex:pageBlock >
            Household Name: <apex:inputText value="{!ID_c}"/>
           <apex:commandButton value="Search" action="{!doSearch}" rerender="pblck"/>
           </apex:pageBlock>
    <apex:pageBlock title="Characteristics Information">
            <apex:pageBlockTable value="{!selctchr}" var="eachpet">
            <apex:column value="{!eachpet.Name}" />
                <apex:column>
   <apex:inputCheckbox value="{!eachpet.Good_with_Cats__c}" />
                    </apex:column>
</apex:pageBlockTable>

       <br/><apex:commandButton value="Search" action="{!dochar}" rerender="sblck"/>
    </apex:pageBlock>
    <apex:outputPanel id="sblck">
    <apex:pageBlock title="Search Results">
    <apex:pageBlockTable value="{!selctpet}" var="eachpet">
        <apex:column value="{!eachpet.Name}"/>
        <apex:column value="{!eachpet.Gender__c}"/>
        <apex:column value="{!eachpet.Height__c}"/>
        <apex:column value="{!eachpet.Name__c}"/>
        <apex:column value="{!eachpet.pet__c}"/>
        <apex:column value="{!eachpet.Pet_Photo__c}"/>
        <apex:column value="{!eachpet.Status__c}"/>
        </apex:pageBlockTable>
    </apex:pageBlock>
    </apex:outputPanel>
       <apex:commandButton value="Request Adoption"/>
   </apex:form>
</apex:page>

public class actionSupportController {
    public List < PCS_Household__c > lstQuery {get;set;}
    public List < PCS_Household__c > OnlstQuery {get;set;}
    public List < Characteristics__c > selctchr {get;set;}
    public string ID_c {get;set;}
    public string test {get;set;}
    public List < Pets_Information__c > selctpet {get;set;}
    Public Boolean Good {get;set;}

    public void dochar() {
        string query = '';
        query = 'SELECT ID,Good_with_Cats__c,Pet_ID__r.Gender__c,Pet_ID__r.Height__c,Pet_ID__r.Name__c,Pet_ID__r.pet__c,Pet_ID__r.Pet_Photo__c,Pet_ID__r.Status__c FROM Characteristics__c where Good_with_Cats__c=:true';
        Characteristics__c obj = database.query(query);
        Good = obj.Good_with_Cats__c;
    }

    public PageReference incrementCounter() {
        string Query = '';
        Query = 'SELECT ID,Name,Household__c,Phone__c,of_Childrens__c,of_Exiting_Pets__c,Address__c,Previous_Dog__c FROM PCS_Household__c where ID=:test';
        system.debug('==Query==' + Query);
        OnlstQuery = Database.query(Query);
        system.debug('==lstQuery==' + OnlstQuery);
        return null;
    }

    public Void doSearch() {
        string Query = '';
        Query = 'SELECT ID,Name,Household__c,Phone__c,of_Childrens__c,of_Exiting_Pets__c,Address__c,Previous_Dog__c FROM PCS_Household__c where Household__c LIKE '%' + ID_c + '%'';
        system.debug('==Query==' + Query);
        lstQuery = Database.query(Query);
        system.debug('==lstQuery==' + lstQuery);
    }

    public void nullify() {
        lstQuery.clear();
    }

    public PageReference nextpage() {
        upsert OnlstQuery;
        return Page.cc;
    }
}

<apex:pageBlockTable value="{!selctchr}" var="eachpet">
       <apex:column value="{!eachpet.Name}" />
       <apex:column>
            <apex:inputCheckbox value="{!eachpet.Good_with_Cats__c}" />
       </apex:column>
</apex:pageBlockTable>

public class actionSupportController {
    public List < PCS_Household__c > lstQuery {get;set;}
    public List < PCS_Household__c > OnlstQuery {get;set;}
    public List < Characteristics__c > selctchr {get;set;}
    public string ID_c {get;set;}
    public string test {get;set;}
    public List < Pets_Information__c > selctpet {get;set;}
    Public Boolean Good {get;set;}

    public actionSupportController ()
    {
          query = 'SELECT ID,Good_with_Cats__c,Pet_ID__r.Gender__c,Pet_ID__r.Height__c,Pet_ID__r.Name__c,Pet_ID__r.pet__c,Pet_ID__r.Pet_Photo__c,Pet_ID__r.Status__c FROM Characteristics__c where Good_with_Cats__c=:true';
        selctchr  = database.query(query);
    }
    public void dochar() {
        string query = '';
        query = 'SELECT ID,Good_with_Cats__c,Pet_ID__r.Gender__c,Pet_ID__r.Height__c,Pet_ID__r.Name__c,Pet_ID__r.pet__c,Pet_ID__r.Pet_Photo__c,Pet_ID__r.Status__c FROM Characteristics__c where Good_with_Cats__c=:true';
        Characteristics__c obj = database.query(query);
        Good = obj.Good_with_Cats__c;
    }

    public PageReference incrementCounter() {
        string Query = '';
        Query = 'SELECT ID,Name,Household__c,Phone__c,of_Childrens__c,of_Exiting_Pets__c,Address__c,Previous_Dog__c FROM PCS_Household__c where ID=:test';
        system.debug('==Query==' + Query);
        OnlstQuery = Database.query(Query);
        system.debug('==lstQuery==' + OnlstQuery);
        return null;
    }

    public Void doSearch() {
        string Query = '';
        Query = 'SELECT ID,Name,Household__c,Phone__c,of_Childrens__c,of_Exiting_Pets__c,Address__c,Previous_Dog__c FROM PCS_Household__c where Household__c LIKE '%' + ID_c + '%'';
        system.debug('==Query==' + Query);
        lstQuery = Database.query(Query);
        system.debug('==lstQuery==' + lstQuery);
    }

    public void nullify() {
        lstQuery.clear();
    }

    public PageReference nextpage() {
        upsert OnlstQuery;
        return Page.cc;
    }
}