Welcome to RootsChat,
In answer to your first question, no, the info on the death certificate was not always accurate. You have a few things you can do here:
1. Check the 1841 census for the family - not the most helpful of censuses (?censi??) but still it would help narrow down a time frame if Dad were dead.
2. Check the 1851 census for the same person - then you only have a 10 year time frame to check if he was alive in 1841 but dead in 1851, maybe only 7 years if you believe he was alive in 1844.
3. Check the marriage certificate - did he sign as a witness - hopefully she did not have a brother with the same name, but bear it in mind.
I think the original parish records will be at London Metropolitan Archives in Northampton Row. Some libraries may also have them on film.
When you do find his death certifiacte, it will not contain his place of birth. You need to find him alive on a census, preferably 1851, to get a rough idea of his birthplace. 1841 data only shows if the person was born in the same county as the census place. 1851 shows the actual town & county. Hope and pray that it isn't Scotland or Ireland - if so, enumerators were often not be bothered with actual towns. Also hope that it was either nearby or wasn't totally incomprehensible and he wrote down what he thought he heard!
Good luck.
Nell