Yes, hadn't.

I find the sentences a bit confusing though. The shift from "her" to "the family" and back again plus the tense in the second sentence (Wasn't "she" the one diffectly affected by the second fateful blow, or was that a third blow?). I thought it might mean:

Before she had even had time to process the loss of her sister, another tragedy struck her family. She had been trying to be there for everyone, always putting their needs before her own, and the inevitable happened.

Ich hätte automatisch "didn't have time" gesagt. Wäre das falsch, oder ginge das auch?

Edith meint, ich solle nicht so doofe Fragen stellen; das würde hier ersichtlich nicht richtig sein.

*redundant*

Might be too late, but:

She hadn't even had time to process the loss of her sister when her family was struck by another misfortune/tragedy/ blow of fate. (Gibson is 100% correct here, dealt a blow of fate is much more elegant than stroke; the standard pairings are "blow of fate" or "stroke of misfortune").

I can't speak for the 'correctness' of the German here, but the 'when' is necessary in English to avoid an unseemly comma splice (as in the OP). I think German is more forgiving about this than English is.

#4, #9: Mir gefällt "process" auch besser als "get over". "Get over" hat hier etwas unangebracht Oberflächliches -- als müsste es nach einer bestimmten Zeit auch mal gut sein mit der Trauer.

#10 I don't much like "process", it sounds a bit formal. I prefer "come to terms with".