PhoneGap app that utilizes CodeIgniter and MySQL?

What you have created is called a Web App.

What you you would like to create with PhoneGap is a Hybrid App.?

PhoneGap/Cordova framework contains a Browser and has plugins that can be added to access Native features which can be accessed through JavaScript.
Native features of the phone like Contacts, Calendar, GeoLocation, Accelormeter etc. Which is not possible when you use a Mobile Browser. Hence the name Hybrid App
(Using the term Browser for simplicity sake. A more appropriate term is Webview container.)
So any web application that uses HTML5, CSS3 and JavaScript will work with PhoneGap/Cordova.
PhoneGap allows you to create Hybrid Apps for all the platforms Android/IOS (Iphone)/Windows/BlackBerry.

Now comming to your questions.
Q1)
You do not need to create any folder.
You just need to point the startup (index.html) file to your Webserver address.

Q2)
As informed earlier your Hybrid app is a like a Browser(with additional features) so the same codeigniter code for securing the form will work.

I hear something about AngularJS. Can I use it for secure form? maybe Codeigniter is not necessary?
Codeigniter is a Serverside scripting tool which is used for processing server side request.
AngularJS is a client side JavaScript framework which is used mostly to create a SPA ( Singe Page Application) web application.
SPA web application relies heavily on AJAX

EDIT 1:
In the index.html file which will be located in the assets\www directory you need to add the following code in the head tag.

<script>    document.addEventListener("deviceready", onDeviceReady, false);     function onDeviceReady() {        // Now safe to use the Codova API     window.location="http://your.website"; }</script>

You also need to tell Cordova that it is safe to load your website in the config.xml properties (you can remove the subdomains part if you don't have any)

<access origin="http://your.website" subdomains="true"/>