I have tried to set-up an email.html and an action.php so that someone can send an email from a website. Here is the code in email.html
Start by checking the return value of mail to see if the message is being accepted by your SMTP server.
Also, per Namecheap's documentation, your From address is invalid.
Only domains that are hosted on our servers can be used in 'From' field. Any domain that is not hosted with us cannot be added to 'From' field. We had to take this measure to prevent sending spam using forums, guest books and contact forms scripts. For your site scripts to work properly you should set 'From' field to email account that has been created in your cPanel.
Even if this were allowed by your hosting company, you shouldn't be sending mail from @gmail.com using non-Gmail servers anyway. It will often be blocked by SPF and other such anti-spam measures on the receiving end.