"I am reading my book."
Translation:Je lis mon livre.
It does, but in this case it would sound like you are saying that you are physically a reading...or something like that.
Lire just goes by itself. Je lis = I read = I am reading.
When it does get accompanied by another verb, it is the verb avoir. "I have read" = "J'ai lu".
You should read this little article on je suis fini v.s. j'ai fini.
And this one on use of auxiliary verbs:
Hope that answers some questions!
The combination "Je suis lis" is not possible, meaning "I am (I) read". French hardly uses the -ing form. It would have sounded: "je suis lisant mon livre" which simply is not used. In general you can translate any verb in the present as a construction with -ing, like Hohenems explained.