It is possible to define my own expression function in live templates?

Absolutely.
Write a plugin basing on the ItelliJ IDEA Community Edition sources.

Key: <liveTemplateMacro implementation="com.intellij.codeInsight.template.macro.CurrentDateMacro"/>

Thank you very much!
Then I have another question: can I get from expression the content of the file, which I call the live template?

yes, Macro interface provides relevant context.