Yes, the display switches on and off. In the top right window, the battery charge, time and date are displayed for a short time.
You can also select under Settings whether the device unlocks itself when the cover is opened. However, this only works if a secure lock screen is not set.

Yes, the function is available

What one should perhaps add is that you can make a phone call when the device is closed.

It depends on what you mean by wake-up or sleep function. As you can read below, the cover is only used to switch the screen on or off. It has no influence on other functions of the device, such as the clock.
Unfortunately, the cover does not work very reliably for me. It occasionally stops working and can only be revived if the cover is removed, cleaned and refitted.

yes, when you close the cover, the screen turns off and vice versa.

In my case, the display switches on when the "lid" is opened. But then it still has to be unlocked. When it is closed again, the screen switches off. In the right-hand corner (recess), you can briefly see the original time, the charging status and the date.