Implications of polynomial time reductions I'm reviewing for finals and have a sample problem that I think I understand, but would like someone to bless my understanding or smack me and tell me why I'm wrong.

I'm presented with a problem $\Pi$  of unknown complexity class.  If I can transform $\Pi$ to some problem $X$, where $X  \in {\sf P}$, what does that tell me about $\Pi$?

I think allows me to conclude that $\Pi \in {\sf P}$, right? If I can reduce $\Pi$ to another problem that's deterministically solvable in polynomial time, and the transformation itself can be done "easily" in polynomial time, then I can conclude that $\Pi$ is deterministically solvable in polynomial time, and therefore that $\Pi \in {\sf P}$ correct?

Conversely, given the same input, transforming $X$ to $\Pi$ in polynomial time allows me to conclude nothing meaningful, since nothing is known about  $\Pi$ right?
Right, if you have an algorithm A that reduces $\Pi$ to $x$ in polytime, and you can solve $x$ is polytime using some algorithm B, then you can solve $\Pi$ in polytime by applying first A and then B.

Using polytime reductions on a problem in P however is rather boring, since you can solve the instance during the reduction and map every instance of $x$ to either a trivial yes or a trivial no answer. Normally you restrict yourself to logspace reductions for problems from P.