This Bible contradiction is from the Skeptic's Annotated Bible.
Yes, it is a sign of a true believer.
Mark 16:17 View context
And these signs shall follow them that believe; In my name shall they cast out devils; they shall speak with new tongues;
No, some who don't follow Jesus can cast out devils too.
Mark 9:38 View context
And John answered him, saying, Master, we saw one casting out devils in thy name, and he followeth not us: and we forbad him, because he followeth not us.
Luke 9:49 View context
And John answered and said, Master, we saw one casting out devils in thy name; and we forbad him, because he followeth not with us.
No, many who have cast out devils in Jesus' name are now in hell.
Matthew 7:22-23 View context
Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works?
And then will I profess unto them, I never knew you: depart from me, ye that work iniquity.