One issue I need to get clear is whether Digital Ocean is giving me port 25

Very simple to test this for yourself, but do note that most virtual host providers block SMTP traffic. Log into your server, and run the netcat command, listening on port 25 (need sudo because the port number is less-than 1024).

steve@cesium:~$ sudo nc -l 25
[download]

Now, on a remote machine, using netcat again, create a connection to the server, and type something, and hit ENTER:

steve@maezi ~ >nc cesium 25
hello, world!
[download]

If port 25 is open on your server, you'll see your typed message on its console:

steve@cesium:~$ sudo nc -l 25
hello, world!
[download]

You can also type something while on the server while the nc command is still running on both machines, and the message will show up on the machine you connected to the server with; in essence, a little chat program ;)