you have to get permission from the designers AND have written proof that they let you use them.
AKA a written contract
this must be posted with your map, and keep in mind, an illegal contract (one that you made not them) will put you in jail and/or get you sued.
Erm. No.
You just need permission. If it isn't granted already (and in Doom's and almost every other game's case it isn't) you need to ask for permission by contacting them. How they grant that to you (in an email, on the phone, on a piece of paper, carved in stone, burned into a piece of wood, on a tape, on a record when played backwards, as a message in a bottle, on a scroll wrapped around a feathered arrow shot into your window at night, tatooed on a baby seal's forehead...) is completely up to them.
Whether they want you to you mention this in your readme (or other conditions) is completely up to them as well.
You don't just get sued out of the blue. You get sued by those who have the rights. If they granted you permission to use their content, they won't sue you for using it. Simple as that.
You also won't go to jail. Most likely you'll have to pay a lump of money if they sue.
The chances to actually get permissions should be pretty slim here. The game is still being sold and although numerous Doom engine ports exist, all the legal ones require you to own the original game before you can use them.
There is no reason why they would allow you to use their content but not the plethora of other Doom related projects out there.
Recreating the textures is your best bet. Morally and legally.
