Yes and yes. Even though the bearings themselves are sealed, it's just to keep the grease in, not to keep the gear oil from leaking. That's done with that inner seal you're probably looking at.
Not sure why it would look like it was all steel though, unless it's just hard and shiny from age. Press your finger down on the inner lip and see if it flexes at all. If so, that's your old seal. If it's fairly hard, that's likely your leak too.
And yes, it should be taken out and a new one put in. Like mortimersnerd said about the bearing, same thing goes for the seal. Even if it wasn't leaking, if it had a lot of miles on it, just replace it while you've got things apart. Since yours IS leaking though, then changing it out is a must.
A common issue when replacing old seals is the new one leaking around the outside. Even though most have that plastic coating on the outside that acts like a glue and sealant, I like to put a bead of silicone sealer around the housing surface before pressing the new seal in. Then rub some clean oil (or gear oil, since it's handy) around the inner lip of the seal before sliding the axle in.
Also check the seal surface of the axle shaft. If it has any groove or wear marks on it at all, dress it up with some super-fine sand paper or emery cloth. Like 400 and up grit.
You want to smooth the surface back down evenly, so that the new seal isn't riding in a worn spot.
Paul