Is every well-founded simplification order a well-partial order?
Every simplification order is indeed a well-partial order because of this simple statement:


  If $R$ is a well-quasi order, and $S$ is a partial order, and $R\subseteq S$, then $S$ is a well-partial order.


Proof: Exercise. Note that this nice property is not true for well-founded orders in general! In this sense, well-quasi orders are much more "stable". It's often the case that a stronger property is better behaved, even if the weaker property is what we want to prove at the end (well-quasi orderdness vs well-foundedness in this case).

Since every simplification order contains the homeomorphic embedding, it is indeed a WPO.

Your more precise statement is correct: we could directly prove that a simplification ordering is a WPO by re-producing the proof for the homeomorphic embedding, but this is not necessary because of the above statement, which shows that it suffices to prove the statement for the "minimal" such ordering.