When you are using the Internet at a place like Starbucks or something like that you are surfing the Internet via a Router that they have installed so several people can surf the internet at the same time. Each person using the Internet is assigned a Private IP Address from the router that Starbucks has installed there. Typically the IP Address of your computer will be something like 192.168.1.x(with x being anything between 2 and 254). This varies from place to place but 192.168.1.x is the most commonly used IP Address numbering scheme. For example when you go to
www.totallyamaha.net, the packet(information) leaves the IP Address of your computer(we'll say it's 192.168.1.2), at that point it is directed to the LAN, inside, or private(whatever you wanna call it) IP Address of the router. At that point the data is then transferred to the WAN, outside, or Public(once again, they all mean the same thing more or less) IP Address. Then the data is tranferred from the router out to the Internet. When the TY server receives the packet from your computer it actually has the IP Address of the router that you are connected to in that packet. Your IP Address is not shown.
So to answer your question in Leman's terms, NO, your IP Address is not shown when you send an e-mail from Starbucks. Now let's say your are connected at home directly to a modem(No router or LAN installed). In that case your IP Address would be attached to the e-mail.
It can be hard to understand without some education about how it all works. If you want you can forward the e-mails to me and I can most likely tell you just by looking at them whether they are real or not. Put "TY" and your name in the subject or something so I know that they're from you.
Anymore questions just ask.