Howdy 
Thank you everyone so far…
In the meantime I gave it up with Windows 10 and I use an old laptop with a running LXQt. The good news, fastboot devices is recognising my FP3.
Next problem:
fastboot flash boot lineage-16.0-20201207-recovery-FP3.img
fastboot: error: Couldn’t parse partition size ‘0x’.
I do not really understand these boot_a/b commands, but with them I have the following result:
fastboot flash boot_a lineage-16.0-20201207-recovery-FP3.img
Sending ‘boot_a’ (65536 KB) OKAY [ 3.267s]
Writing ‘boot_a’ FAILED (remote: ‘Partition flashing is not allowed’)
fastboot: error: Command failed
In the meantime I’m googeling, hopefully you folks have also an idea…
EDIT1: Realized, that I have to fastboot oem unlock… Then I was kind of happy, that I received new steps, but then… weird… it rebooted 3 times and now I have the "standard preinstalled system again, so I have to repeat all the steps again?!? :-S I keep you all updated 