I have a script here that  I found and is almost perfect for what I need.. although when pressing the button "add competition" i need it to be added to a different object "Test Competitors" which is a related list shown in page of  "Account" (related list).
Could anybody "tweak"/ adjust this script accordingly!? ]

public class AccountSelectClassController{
 
    //Our collection of the class/wrapper objects wrapAccount
    public List<wrapAccount> wrapAccountList {get; set;}
    public List<Account> selectedAccounts{get;set;}
 
    public AccountSelectClassController(){
        if(wrapAccountList == null) {
            wrapAccountList = new List<wrapAccount>();
            for(Account a: [select Id, Name,BillingState, Website, Phone from Account limit 50]) {
                // As each Account is processed we create a new wrapAccount object and add it to the wrapAccountList
                wrapAccountList.add(new wrapAccount(a));
            }
        }
    }
 
    public void processSelected() {
    selectedAccounts = new List<Account>();
 
        for(wrapAccount wrapAccountObj : wrapAccountList) {
            if(wrapAccountObj.selected == true) {
                selectedAccounts.add(wrapAccountObj.acc);
            }
        }
    }



<apex:page controller="AccountSelectClassController" sidebar="false">
    <script type="text/javascript">
        function selectAllCheckboxes(obj,receivedInputID){
            var inputCheckBox = document.getElementsByTagName("input");
            for(var i=0; i<inputCheckBox.length; i++){
                if(inputCheckBox[i].id.indexOf(receivedInputID)!=-1){
                    inputCheckBox[i].checked = obj.checked;
                }
            }
        }
    </script>
    <apex:form >
        <apex:pageBlock >
            <apex:pageBlockButtons >
                <apex:commandButton value="Add to Competition" action="{!processSelected}" rerender="table2"/>
            </apex:pageBlockButtons>
 
            <apex:pageblockSection title="All Accounts" collapsible="false" columns="2">
 
                <apex:pageBlockTable value="{!wrapAccountList}" var="accWrap" id="table" title="All Accounts">
                    <apex:column >
                        <apex:facet name="header">
                            <apex:inputCheckbox onclick="selectAllCheckboxes(this,'inputId')"/>
                        </apex:facet>
                        <apex:inputCheckbox value="{!accWrap.selected}" id="inputId"/>
                    </apex:column>
                    <apex:column value="{!accWrap.acc.Name}" />
                    <apex:column value="{!accWrap.acc.BillingState}" />
                    <apex:column value="{!accWrap.acc.Phone}" />
                </apex:pageBlockTable>
 
                <apex:pageBlockTable value="{!selectedAccounts}" var="c" id="table2" title="Selected Accounts">
                    <apex:column value="{!c.Name}" headerValue="Account Name"/>
                    <apex:column value="{!c.BillingState}" headerValue="Billing State"/>
                    <apex:column value="{!c.Phone}" headerValue="Phone"/>
                </apex:pageBlockTable>
 
            </apex:pageblockSection>
        </apex:pageBlock>
    </apex:form>
 
</apex:page> 


Also I need this VF page to be popping up when clicking "new test competition" on the related list as picture below
[User-added image]

If need anymore info please let me know.. I really appreciate the help!!

Thanks Vincent