The first two sentences are correct, I suppose. You could use either the past simple or the past perfect. Both sentences make sense. I prefer the second one ' I lost the umbrella I had bought the day before', but I'm sure many people would say 'I lost the umbrella I bought the day before'. But then I'm old-fashioned and I still 'feel' that when you speak in the past and want to go to a previous past, you should use the past perfect.
There is a difference between the second pair of sentences.
'I /He knew that my mother was a nurse' means that your mother was (and may still be) a nurse at some time and 'He' knew it at the same time.
'He knew that my mother had been a nurse' means that He knew (at some point in the past) that your mother had been a nurse sometime before then, not at the same time.