One, they can't just openly state the Titans' real name, because though everyone knows them (Robin/Dick, Raven, Star/Kori, Cy/Vic, BB/Gar, etc.), there are people who disagree like Rowen for instance, who thinks Robin is Tim.
Hints are given at their real names (ex: for Robin: Fractured, How Long is Forever, Haunted), but they will probably never be truly given (I might be wrong), because the creators want people to know them as heroes and not the people behind said heroes.
As for the money, comic wise, it is said that Cy's father before he died, gave Cy the money to purchase/build a house and Cy had the Tower constructed. Either that or Bruce funds the whole thing, BB spent his hidden fortune on it, or the citizens of Jump gave to the Titans free of charge. However, it happened is irrelevant, especially considering all the inconsietancies with this show.
But, we know full well that Slack does follow the comics somewhat (ex: Season 4, Terror of Trigon; Season 2: Judas Contract), but just not fully, because he wants the show to be all his own...