You described it to a "T". This kid is the bastard child, hidden from his dad and the public for years (or so I read). Then one day the secret is out and for a while he is like radioactive. Glad to see Arnold is treating him like a family member now. But the desire to prove himself is probably nuclear hot in this kid.
No, Arnold knew from the beginning that he was the father. Arnold support them financially from the beginning. The kid may not have known Arnold was his father, but he grew up as part of the extended family. Arnold's other kids all knew him. That's why they were so upset when they found out that he was their half brother.
Now Joseph didn't hit puberty and suddenly look like his father. I'm sure many people noticed the resemblance very early on.
Also, the mother is not Mexican, but Guatemalan. Big difference, especially to Mexicans and Guatemalans.