Hi Jamie!
Just some ideas off the top of my head -- please tell me which of them you have already considered and disproved:
The indication of being a widow on the census: I've seen ladies call themselves widows when their husband left them. Maybe he came back, and fathered more children! Also, census takers are not perfect either. Do you have any other information about the death of Edward? Looks like in 1871 she didn't say she was a widow.
The ages of Joseph's sibs: Are these birth dates from census records or from more contemporary sources? It is not uncommon for people to only age 6 to 8 years between enumerations, even though the enumerations are 10 years apart. Time flies! Happens to us too, even though we are much more aware of our birthdate for official purposes than people were back then. And then there are all the other possible reasons for wishing to appear younger than really is the case. I've seen instances where this results in losing at least a decade, which would put those sibs back into the pre-widow time frame maybe...
The birth and marriage certificate info: Any chance these are late registrations? In Canada, there was a whole wave of late registrations when the paperwork was required for purposes that didn't exist when the event happened. Then you have people trying to remember things that happened long ago -- amazingly, a lot of it turns out to be right! Not all of it...
Are you looking at original docs for birth and marriage registrations? Transcribers get tired and make mistakes too.
When i run into things like this situation you are facing, i try to hypothesize circumstances that could result in the records being what they are, and then try to prove/disprove the hypotheses. One is that the registrar forgot to record Joseph's father's name, and decided to assume he was named after his father. Or Joseph himself can't remember meeting his dad, so decides or is advised to assume he was named after his dad. Or, his dad really was Joseph somebody, but his mother's last name was still Sandell, so some more assumptions and delicate coverups ensued...or after Edward passed away, Betsy married his brother or cousin Joseph, who also passed away or went away before the next census...
Is anyone doing any DNA in this line? It is recent enough that you could expect some clues to emerge.
Let us know what you find along this interesting journey!
Cheers
Shirlea
