The birth death adjustment has nothing to do with births or deaths of people. As the reference says it is to take account of the start and termination of primarily small businesses. The business survey does not include new businesses or those that fail before they are part of the survey. The catch is that the adjustment tends to add more jobs during a downturn, that is it lags the change.
As to the other point while job losses were 216000 it seems that the number employed dropped by 981,000, according to the household survey, while the unemployed dropped by 387,000. The last would be due to discouraged workers no longer seeking work so not being counted, and if my memory is correct there is a time limit on the period one is considered unemployed.
The numbers don't match well. Part of the problem is that the employed includes those working part time who want full time work, and those who have had to take much lower paid jobs. However both have diminished spending power which is not fully reflected in the unemployment rate. See
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