Let's say, for the sake of argument, I join the Brotherhood of Steel, therefore I get Danse as an available companion. I then, later, decide to join and side with the Railroad. If I remember correctly, as part of that I betray the Brotherhood so would I loose Danse as a companion? I know it seems obvious I will but I wanted to check.

In addition, do I retain him as a companion after I find out his secret, assuming I keep him alive?

Yes, faction exclusive characters are only available as long as you remain friendly with that faction, although Danse will not shoot you and would just ignore you once his personal quest is completed. Danse can be used after the personal quest, you will get his power armor, and he will get an upgrade. He will become one of your most loyal companion in the game after that quest, at high relationship, he won't ever turn on you, even if you destroy Goodneighbor, Diamond City, or slaughter all the settlers, help you kill BoS members, and give you 20% damages increase against most popular enemies in the game. After his personal quest, the BoS will turn hostile toward you and Danse as long as you bring him along, but Danse will help you kill the BoS members.