Trying to answer both replies here, so if I miss something, forgive me.
If the other men are also the children of your Rhoda Pearce, then it could be that the people they were apprenticed to were in Washington County, but then left. Or, Rhoda might have moved around that three county area, in which case, you might find mention of her in either the bastardy bonds, apprentice bonds, or Court of Please and Quarter Sessions records for Beaufort and Martin. But, the apprentice bonds won't give you information about the father(outside of whether the child was considered mulatto or colored, which meant anything from having any African blood to having Native American blood), and unless Rhoda named the father, a bastardy bond won't give you information on the father, either. The court records would be more likely to give more information, but even then, the court would be focusing on the mother, wanting the bond money from someone. If she didn't name the father, then someone else would have been expected to pay the bond, either the male head of the house where she lived, or she'd have to put it up herself.
It's good that you've ruled out the one Rhoda Pierce, and if James Henry Sparrow was a half-brother to David, then you know that as of 1873, Rhoda is deceased (at least as far as James knew). Family lore, from your family, the people looking for Washington Keys, and perhaps the families, if any, of Daniel and Robert, might be your best leads. The fact that David never mentioned siblings isn't that surprising, given how young he was when he was apprenticed to James Magee, as he wouldn't have been around the others, who are all younger than him. I do wonder though if Daniel and Robert were full siblings, and not half-siblings (provided of course, that this is David's family). And there's also the two Pearce girls at the hotel in 1860 - relatives through Rhoda, siblings/half-siblings, or just a coincidence that they had the same surname?
I go down to Washington County once a month, and sometimes I have the opportunity to go to the courthouse. If you need/want something looked up, let me know. I've had times when I needed to go to Beaufort County, but right now, I've gotten just about all of the local records from there that i need. The court records are up at the NC Archives. I do collect court records, but right now I'm still gathering Tyrrell County records, as I've got just all of what they have for Washington County at the archives (although I still need to see what they've got on microfilm). My regular e-mail is email@example.com
, if you have something you'd like looked up and don't want to go into it here.