Yes, sleep information from the Tempur-Pedic® Sleeptracker® app is shared with the Apple Health app if you enable this option during the initial setup flow.
The Tempur-Pedic® Sleeptracker® app writes sleep data to the Apple Health app at the "max resolution" Apple supports, by logging periods you are in bed and periods you are asleep.
Summary data is displayed in the Health app by selecting Sleep > This Week > Show All Data.
Note that currently Apple doesn’t support logging sleep states to Apple Health such as light/deep/REM
Heart Rate and Breathing Rate:
The Tempur-Pedic® Sleeptracker® app writes all data to Apple Health (at 5 minute resolution) throughout the night.
Summary values are viewed in the Health app by selecting “Vitals” from the home Health app page.
To change what information is shared between the Tempur-Pedic® Sleeptracker® app and Apple Health, go to your iPhone’s settings under Privacy then select Health > Sleeptracker®.