Puzzles 1-3 were useful in clarifying my understanding (thanks John) but I will leave the answers to others. For Puzzle 4, my go-to poset example is always the set of pairs \\( (x,y) \in \mathbb R^2 \\) where \\( a \leq b \\) when \\( a_x \leq b_x \\) and \\( a_y \leq b_y \\).
I guess a good "Fong-Spivak poset" example is triples \\( (x,y,z) \\) with the same ordering as above (\\(z\\) is ignored). All triples with the same \\(x\\) and \\(y\\) but different \\(z\\) are equivalent but are not equal.
[Edited to see if the new MathJax recipe works - it does!]