No one can or should predict the outcome of a custody battle for you. I would suggest, if you want to get some control over this situation that you file for paternity and get (apply for) the orders you want. From what you've described it does not sound like a 50% custody arrangement will be practical. When making his or her decision, the court will consider what is in the best interests of the child keeping in mind the goal of frequent and continuing contact with both parents. Dad's military commitments would be factored into that decision in light of his frequent travel. Further, if you file for paternity, you can bring up the issue of the girlfriend and your discomfort with her habit of showering with your daughter.
Hire a lawyer.
You need to speak to an experienced family law attorney, to get yourself a BINDING agreement for custody, and an ENFORCEABLE order for child support. It is UNLIKELY that a court is going to permit a 50/50 time-share with someone being re-deployed, whether or not you think his girlfriend did something inappropriate. BUT, without much more specific detail, a court is NOT going to prevent Dad from having any visitation at all based on your dislike of his girlfriend.
I agree with the previous answers. You cannot predict the custody orders, but it is best to get binding and enforceable orders from the Court. The Court will do what is in the best interests of the child, so if your ex is getting deployed, has a drinking issue or criminal history, that would mitigate against 50/50 custody for your ex. But you should definitely hire legal counsel
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