They are called 真田紐(さなだ ひも). Their width is usually measured in 分, where 1 分 ≅ 3 mm.
It's nice if the ends of the cords are tied as well. This is how to do it.
Lids
They can warp with time. Since the writing is quite important, do not spray them to straighten. Instead, leave in a moist room (bathroom after a hot shower) for 60' with the bent side facing up, towards the moist air.
Then put a weight on it, making sure the writing is protected