Big EDIT :
Hi,
In between I found a dell optiplex 390 MT for really cheap.
I was able to move my 3010’s CPU/RAM/HDDs into it and it worked at first try (at least it is usable again).
The boards are identical, only the bios and rom chip seems to differs ( 4MB > 8MB - 3206E > 6406E).
It may be as simple as to dump the 390 bios and flash it to a FF filled 3010 bios chip.
I also found a 3010 bios dump over internet, that I may try first (I do not really want to desolder the 390 bios chip for now).
Or, is there a program that can fully back-up the optiplex 390 bios without the need for a programmer ?
By saying “fully” I mean : would the file be directly usable with a programmer ?
Many thanks again !
@pass-by - BCDboot/BCDedit cannot affect the BIOS, but a faulty hard drive may affecting trying to boot. Remove your HDD/SSD (ALL) and see if you can then access BIOS or not.
What is your BIOS chip ID? And, what software/version are you using to dump the chip? If you are not using correct software or version for some chip ID’s, then read or write can fail (some only minor corruption, others major corruption/failed dump etc)
Hopefully you did not erase or write to chip yet! 
Ohh, I see you did already, bummer!! Hopefully your dump is not too badly corrupted 
Please upload your BIOS chip dump by itself for me, I do not need the other files and am on limited internet.
Please also include in this, images of ANY and ALL stickers you see on the motherboard, be sure to check top/bottom/side of LAN port metal block, side of 24pin connector, top/bottom sides of PCIE slots, and back of board.
And if you know it, please send text of LAN MAC ID. If you are not sure, maybe you can check router logs and identify it from there. If not, it should be on some sticker anyway, so hopefully it will be included one way or another 
Big edit :
Not needed anymore.