is it possible when you book time at ‘real work allocation‘ for the first time for an activity that the status will automatically be set to ‘in progress’?

Yes, this is default behavior. You'll see a small green flag that will inform you about the change.
This automation can be disabled in global parameters screen (tab "work")

Because normally if you have no more time left, the status will be set with the last booking to ‘completed’.

This is also available, and set as default behavior. You see a "finish" flag, but it may be stiked if move is impossible (for instance if result is required and not set)
This automation can also be disabled in global parameters screen (tab "work")

And we also have one resource, a project service who has also to book time for some projects. If there is no more time left or he set the left time to 0 the activity gets not automatically to ‘completed’ but to ‘cancelled’.

Because in your workflow, it is the fist status for the corresponding profile that will check item as "done"

Thanks for your help.

is it possible when you book time at ‘real work allocation‘ for the first time for an activity that the status will automatically be set to ‘in progress’?
Yes, this is default behavior. You'll see a small green flag that will inform you about the change.
This automation can be disabled in global parameters screen (tab "work")
This works now. But there is no check in the box and date of handled. Is this a bug? Or is this normal?

It depends on configuration of your status.
The status must "check handled"

we have in the status checked handled

and also set in the global parameter the "set to first handled / done status" to yes

But still there is no automatically check in the box at handled or done if we book time at ‘real work allocation‘. And also no date will be set.

On activity type, set handled as locked to ensure it is automatically set from status and never manually changed.

Can you tell me, if you mean this one:

Thanks you.

Yes