I don't like to run skype all the time anymore. Is there any way to get chat notifications sent to my email account?
The way Skype works is that when you're offline, you are offline. This means you will be able to receive messages only when you and the partner are online at the same time. Vice versa, if you are offline, you'll only get a notification when you sign back in and your partner is or becomes available too.
You might as well stay online all the time.
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
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.)