Is this made for a 8 tablespoon (.4 oz) butter stick - the kind of butter stick that comes four to a box?

It can use those but it is better with the wider 4 oz butter sticks, like those as Trader Joes. But we have used both with success

Is this container meant for Western Stubbies or Elgin size butters?

I am not familiar with Stubbies or Elgin. The container works great if you use 1/2 of a quarter pound stick from a 1 pound package of butter sold divide in quarter pound sticks . You can always cut the butter to fit the container.

Is there a lid for this, so you can store butter without it being visible?

If that's a cover, it doesn't appear to completely "cover" the butter. I can see the butter peeking out the side of the bottom.

Does this have a lid, in order to store butter without it being visible?

The butter spreader has a plunger that pushes down on the butter stick into the plastic body and a cap for the end of the plastic body. When finished, cap the end of the plastic body and the plunger is the cap at the other end of the body. I like it for buttering corn on the cob.