SMS messages, as any other form of evidence, may be admissible if relevant to assist a judge a matter pending before the court. Assuming you will be able to lay foundational grounds for introducing them in court, the court may admit some of them if it determines that they are relevant and otherwise admissible.
Not to show that your ex did anything , not any more than if GF had told you this stuff face to face, but isn't in court to testify.
You need to speak to an experienced family law attorney NOW. Unlike a money issue, if the court makes the wrong call on a custody issue because the evidence didn't get put on correctly, it's difficult to go back and "fix" the mistake and its effects on your child.