Most of the emperors and "nobles" (always a problematic term when it comes to the Roman state) that you're thinking of adopted because they lacked natural heirs. Scipio the Elder (Africanus' son), Caesar, Augustus, Tiberius, Nerva, Trajan, and Hadrian (to name just a few) all lacked surviving male heirs by the times of their deaths. Vespasian, Antoninus Pius, and Marcus Aurelius, to name only emperors from before the Third Century, when things got messy, all were succeeded by their sons. The Romans had no problem with primogeniture--under normal circumstances it was the son who would inherit

That said, legally, since Romans had the right to make wills, the sons of a man might not necessarily legally inherit any of his property, only his name--and even then you could be disowned. But that wasn't usual at all, and in all but the strangest cases the eldest son inherited the lion's share of the estate (e.g. Cicero and his brother Quintus, who inherited next to nothing and had to actually buy some of his lands off his brother, or the Gracchi)