On the email question:

You can sent alerts TO gmail (or anywhere else) by simply filling in 'youraddress@gmail.com' in the Email Address to Send Picture To: field when configuring alerts in LogiSphere. Gmail does not provide SMTP relay services (you can only send outbound mail via their Web interface) so you need to use your ISP's SMTP server in the Mail Server: field. If you are not sure what the name of your ISP's server is, take a look in the configs of your "regular email" program (Outlook or whatever).

On the question of remote access:

I can't even begin to answer properly without more information. Most problems seem to come from running LogiSphere behind a router. Running behind routers is fine, but you must not forget to open a port on the router to expose LogiSphere to the Internet. LogiSphere listens on Port 8080 by default, so if you haven't changed the defaults just config the router to forward Port 8080 to the machine running LogiSphere.

Router configs are the most common cause of trouble but not the only possible cause. Post more about your setup if you need more help. What kind of router do you use? What kind of Internet connection do you have? And so on.