Was Dougie LeClair actually Doug's son?
Yes, Dougie is Doug's biological son.
After Addie (Julie, Steven, and Hope's mother) died, Doug wanted another child. He decided to have one with a surrogate. Then he and Julie reunited and he figured they'd have a child at some point, so he called off the surrogacy plans, unaware a surrogate had already conceived.
He had a pregnant housekeeper and had no idea she was the surrogate. Her name was Rebecca North. Rebecca had no idea Doug was the sperm donor she was supposed to be carrying a baby for. Robert LeClair loved Rebecca. She liked him, but wasn't in love with him, although she did agree to marry him when he asked, to give her child a name. They named the baby after Robert's best friend, Doug, and asked him to be the godfather.
At some point, Doug found out little Dougie was his son, but it was all kept quiet. Later, Rebecca and Robert divorced, and she reconnected with an old boyfriend and married him. They left town with little Dougie. Then they were killed in a accident. Robert went to get Dougie, since he was his legal father. They moved to France and we never saw little Dougie again. Supposedly he renamed the child Charles.
Dougie had a Horton ornament since the Hortons all loved him. It was identical to Hope's. It appears when Hope's ornament broke, the prop department just painted her name over Dougie's ornament, back in the day. So, Dougie was "ornament worthy" when he was in Salem.
Here is Hope with Dougie:
Here's Dougie's ornament before it was repainted with Hope's name. Hope's current ornament was Dougie's: