Well, the very first thing to do is to get a paternity test, the Court would order it. Then, if he really is the father, child support and custody/visitation would be worked on. I get that he lives in Mexico, but I don't see why he wouldn't be allowed to come to the US for short periods of time to see his daughter. Also, if the issue was resolved legally, Dulce would most likely be allowed to visit her father in Mexico.
Her means of support might have came from her current boyfriend and father of the baby that she's pregnant with, he was probably working. Now that he's in ICE's custody, I'm not so sure how she's holding up. I hope her parents are taking care of her.
My understanding is that, once he realised that Dulce was his daughter, he wanted to finish his education, and that's why he went to Mexico to attend University. It seems like he wanted to get things set legally but also wanted to get his life together first by finishing his education and maybe getting a job.
What do you mean by "legal right"? People can and do visit the US everyday from all around the world. Dulce's father could simply enter the US to visit his daughter, spend a few days and go back to Mexico. Assuming he he's the father (which we don't know for sure, but I choose to take the mom's word for it), he still has the right to spend time with his child, even if he doesn't live in the same country. Lots of families live with that reality and, once kids get a bit older, they even visit the parent who is abroad. It happens.
Don't go there, we don't know how old he is. From the picture I saw, he doesn't look much older than Dulce's mother, they were most likely the same age or around the same age.