These are good questions. And much discussed of course.
There are a number of quite different ways of answering the question. I will mention the approaches I favour.
A key question to focus on is whether we can think of the universe as a closed or open system.
You might argue that the universe winked into being out of nothing (perhaps a quantum "nothingness") and so it is an isolate system. There is no need for entropy outside as there is no outside. The same goes for a cyclic universe model where re-collapse somehow recreates the intial low entropy condition.
But instead, a different model would be to think of the universe as a dissipative structure, an entropy degrader. So it would be something that arises by disposing of a greater gradient "outside" itself - just like a cell or dust devil is an orderly structure that arises by the disposal of entropy.
You have to ask how does an expanding universe fit into the standard closed system approach. A universe that was static, could not cool. The fact that the universe expands (and so creates a heat sink within itself) is what allows the disposal of entropy.
So thinking about this, we in fact have a hybrid answer suggesting itself.
The universe is a closed, and even properly isolated, system of some kind. From the evidence of the big bang (excluding dark energy perhaps), all its "contents" in the form of heat/light/energy fields were there from the start. The first law of conservation applies. But it is also an open system in that it expands and is making a giant heat sink, in effect transporting all that energy density "bitten off" at the big bang moment across the boundary of a structure (the structure of an expanding, cooling, spacetime of certain dimensional symmetries) and creating a vast internal heat sink.
The normal open system transports (n)egentropy from a source on one side to a sink on the other (like a boiling pot with a flame to one side, colder air to the other). But the universe is not moving anything outside, just spreading it around to create an inside.
So entropy concepts can be applied. But we have to then think more radically about the kind of structure that is doing the entropification.
Is it a closed system? Is it an open system? Or is it something that combines aspects of both in a way we are not used to thinking of things.