Hi Tao,
I did not have ADXL313 in hand to double confirm your condition. But I have tried other similar part ADXL362, below is the test result.
1. DATA_READY interrupt can be generated in sleep mode. Of course, in normal measurement mode, the interrupt will be generated too.
2. You can read STATUS register to check whether the data is ready.
3. You can use watermark interrupt in FIFO mode.
BR
Neil