Driving a 14-pin with DFB laser using D2570H
we don't understand the principle of the bias tee connected to the cathode of the laserdiode. How does it work?

You're right that this isn't the typical bias tee arrangement.
But if you ground pins 11 and 13 (the laser anode) and bias pin 3 below ground to draw your desired DC laser current, then you can apply an RF signal at pin 12 and the inductor on the pin 13 branch will prevent the RF signal being shunted to the bias circuit. Similarly, the 20-ohm resistor on the pin 12 branch will improve the matching of your RF feed's characteristic impedance to the laser (which is likely fairly low impedance).
Likely you will want to use an external DC-blocking capacitor in series with pin 12. And you may also want to include a second external inductor in series with pin 3, particularly if you want to use broadband modulation (if your RF signal band includes substantial low-frequency components). You'll want to do some modelling, including device parasitics, to determine the best choice of external components for the modulation scheme you are using.

the datasheet mentioned that pin 11 and pin 13 should be grounded, but should pin 8 to 10 also be grounded since it doesn't mentioned it?

Pins 11 and 13 are the signal ground. The return current for your RF signal will appear on these pins (maybe mostly on pin 13). You should provide a low-impedance connection between these pins and the RF return path.
Pins 8, 9, and 10 are the case ground. If the case is exposed to the outside world, these should be tied to your chassis or enclosure. The ideal way to connect chassis and circuit grounds is probably worth a few questions on its own.
If you're just making a demo circuit (no enclosure, no concern about ESD or EMC) you could tie these to signal ground, or if you have a semi-isolated RF ground, tie them to the main circuit ground instead of the RF ground.
Note
Be aware that these lasers are probably quite sensitive to ESD, and always use ESD precautions (wrist straps, smocks, heel straps) when handling them. To be honest I'm somewhat impressed, given that Agere has been out of business for over 10 years, that your instructor still had some of these devices in working order if they've been used in a teaching environment.