Yes, your proof is correct.

This is correct. By definition, A∪B ⊆ C∪D means that for every x, if x ∈ A∪B, then x ∈ C∪D. Any proof of the statement "For every x, if P(x) then Q(x)" starts with: "Fix any x and assume P(x)".(this is where I'm unsure. Am I allowed to state this?)

Let x∈ A∪B (by hypothesis?)

It should say, if x ∈ A∪B, then x ∈ C∪D.By definition of a union, if A∪B then C∪D.