to test if port 25 and 587 is open is pretty simple, just get a linux live CD, fire it up and enter:
telnet smtp.gmail.com 25
Trying 74.125.206.109...
Connected to gmail-smtp-msa.l.google.com.
Escape character is '^]'.
220 smtp.gmail.com ESMTP s184sm5768906wmb.11 - gsmtp
or for port 587:
telnet smtp.gmail.com 587
Trying 74.125.206.109...
Connected to gmail-smtp-msa.l.google.com.
Escape character is '^]'.
220 smtp.gmail.com ESMTP ml1sm25852287wjb.46 - gsmtp
quit
221 2.0.0 closing connection ml1sm25852287wjb.46 - gsmtp
if you see this above, you can reach the smtp server from your LAN where the camera is
telnet smtp.gmail.com 25
Trying 74.125.206.109...
Connected to gmail-smtp-msa.l.google.com.
Escape character is '^]'.
220 smtp.gmail.com ESMTP s184sm5768906wmb.11 - gsmtp
or for port 587:
telnet smtp.gmail.com 587
Trying 74.125.206.109...
Connected to gmail-smtp-msa.l.google.com.
Escape character is '^]'.
220 smtp.gmail.com ESMTP ml1sm25852287wjb.46 - gsmtp
quit
221 2.0.0 closing connection ml1sm25852287wjb.46 - gsmtp
if you see this above, you can reach the smtp server from your LAN where the camera is