"The value of the household production is hard to measure. But can I draw the conclusion that the value of real GDP plus household production is lower in the recession than in the expansion because the standard of living is lower?"
That's not clear. If I produce something in my home to consume (such as a home cooked meal), it clearly increases my standard of living, yet the value added by cooking my own meal is excluded from GDP. In a recession, if consumers are more inclined to consume household produced goods, then measured GDP will systematically exaggerate the negative effects of recession. This is plausible, especially if you think about the effect of recessions on the service sector (restaurants and entertainment in particular).