EDIT: Just noticed the pages of help after I actually replied. I just woke up.
I've never seen the fastboot words come up on my device, and haven't dealt with it with others, so Misledz, could you please help here?
If you can read this it was sent using my GT-P7510.
I finally managed to borrow a cable from a friend of mine.
So I plugged the tab in downloading mode to the usb...and Tab detected and drivers installed!!!!!!!
I finally get the communication, and restored the Tab.
Really strange that the cable allows to charge but not to communicate?! Plus, with the defect cable, the battery is charging when I plug the usb, with the good one, it doesn't?!
Whatever, thank you all for your time and advices.
PS : do I have to put a "resolved status" for this topic or not?