Contact Form plugin from www.123ContactForm.com
Make sure you have unzipped the folder in the root of the /wp-content/plugins/ directory.
The most effective way is using the API key.
No, you don't have to paste the code again.
Yes, your forms should be visible and working as before, as long as you have the latest version of 123ContactForm plugin. Still, some users reported that after updating the WP version, their forms got broken. So, please check your forms and in case they're broken, all you have to do is to republish them.
Please see author's blog.