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!]