Hi! 
You probably already did this, but just to make sure: Did you also try to get into Fastboot mode? You can do that by pressing power + volume down for a couple of seconds, then the Fairphone logo should pop up and the LED should light up blue. If that works, you can install OpenOS to get Android running. AFAIK OpenOS also includes TWRP, which you could then use to install Lineage (there is probably an easier way to do that, but I did that several times and it workes). Although there probably isn’t much relevant data on the phone since you’ve already used another one, keep in mind that this will delete your entire storage.
If you can’t get into fastboot and you already tried charging the phone for several hours without any response and plugging the phone in and inserting the battery afterwards (also have a look at the battery guide and #dic:kickstart), you could try contacting a Fairphone Angel in your area (the closest ones to Lille are in Paris and Antwerp)
To answer your last question: If you bought the phone less than two years ago, yes you are, as well as if you bought the battery less than one year ago and it is a battery-related issue.
I hope I could at least help you a little. If you have anymore questions, feel free to ask ^^