As long as the address you are sending the email to is on that server smpt should allow you to send it an email, thats how smpt works, your just doing it manually.
What it should through an error on is if you want to send email to a recipient on another server, you should get denied.
"... I expected that the IP of the computer connecting to the server in order to send the mail would have to be from the domain that it was actually claiming to be sending mail from."
This can be done, but it has to be set up correctly. Some server check for the PTR records, thats what most ISP do. I've never set that up for postfix but you could use Postfix's reject_unknown_client option, which will trigger if the reverse and forward DNS don't match, which is the only way to correctly identify a client
name. There is no built-in option to only check for a PTR record that i know of.
you might find this useful: http://www.postfix.org/postconf.5.htm