Can average labor productivity fall even though total output is rising? Can the unemployment rate rise even though total output is rising?

Yes; Yes

Explanation:

Average labour productivity can fall even though total output is rising as long as employment is increasing. When employment increases, the total output increases, but it can lead to a decline in average labour productivity.

Average labour productivity / Total output

Similarity, it is possible; if the average productivity increases the unemployment rate can decrease even though the output is rising.