1. You must connect either to your network or to the internet from a publicly accessible Wi-Fi zone such as in a hotel, restaurant, Starbucks coffee shop, internet shop, airport, etc. The site must be open and allow anyone to connect wirelessly to the site through which you can access the internet. You would then access your mail using a browser provided your email provider supports webmail access.
2. Your Verizon service is distributed through your home network either wirelessly from the modem itself or via a wired/wireless router that you provide. If the router provides both wireless and wired connections then your Touch can connect wirelessly through the router to access the internet.
3. See Answer 1 above. It is possible to connect to your home system via the internet to access your home computer's resources using a VPN connection. You would need to run a VPN server on your home computer. Then you need to access the internet through a Wi-Fi connection remotely.
Here are some KB references:
iPhone and iPod touch- Tips when using Wi-Fi
iPhone and iPod touch- Mail Settings
iPhone and iPod touch- Email Account Setup
iPhone and iPod touch- Setting up VPN
iPod Features Guide
There is a nifty utility for activating a VPN server on your home computer called iVPN -
VersionTracker or
MacUpdate.