1. Can't be done from shortcode. Shortcode is executed when it is encountered in content. By then, the head has been sent and closed, and the body opened. Do what you need in head from your plugin.
2. See Pages#Creating_Your_Own_Page_Templates. Assuming your frontend editor works the same as the backend.
3. Makes sense to me to use your theme's methods and templates to display posts. Even if it's not the "cleanest".
4. You don't have to use WP functions, but they can make things easier. You might consider using javascript or JQuery to check the fields before posting the form, could reduce server requests. You still need to re-verify all posted data at the server since the user may have compromised the client side script.
An ambitious newbie coding project! Good luck.
Never forget that any user submitted data can be malicious. Be careful!