Can I make my email address visible in the "Contact" section of my profile?

No. This is designed as a means of security, to protect your address from spammers, and is why the profile email sending function (the little envelope at the top of your profile) is used. If you choose to publicly post your email address, you may do so for example in the "About me" section or CV of your profile; please be aware of the risks of doing so, however.