I don't like to run skype all the time anymore. Is there any way to get chat notifications sent to my email account?
You may run Skype "headless" on any Linux server or workstation you have access to and have Skype send each incoming text message via email.
The headless setup is attained using Xvfb X virtual framebuffer. This way your instance of Skype will run completely in background, perhaps on a remote machine, without actually showing anything on any graphics display.
Re-sending Skype messages via email is done with the Linux Skype's "Execute script on Notifications" feature.
This feature runs any command of your choice with two possible parameters: sender name or Skype id and message text. A working example of such command line could be
echo "%smessage" | /usr/bin/mail -s "Skype message from %sname" firstname.lastname@example.org
You can also register another Skype account to run normally and forward voice calls to it for free from your main account running headless.
(There is also a way to forward Skype voice calls from the headless Skype to a SIP server for free, but that's another story altogether.)