Yes. You can use Contact Form 7 plugin to create a form that will send messages to your email (like gmail, etc) and use another plugin from the same author, Flamingo, to have those messages saved in WordPress database and available in the dashboard.

Yes. You can use Contact Form 7 plugin to create a form that will send messages to your email (like gmail, etc) and use another plugin from the same author, Flamingo, to have those messages saved in WordPress database and available in the dashboard.

If you prefer to avoid plugins, here's what you need:

1. Register non-public custom post type message to store submissions
2. Create custom html for your form (probably with js validation)
3. Intercept form submission on init hook
4. Validate and secure input
5. Create new post of message post type with wp_insert_post()
6. Send email with wp_mail()
7. Output results below the form (validation errors or success message)

As you see, plenty of work/code. If you're a developer, you should go this way to tailor it exactly for your needs. If not, I would strongly recommend you to use mentioned plugins (or any good alternatives) to avoid having some troubles with insecure form.