Ahhh… I also commute from Mt. Lebanon to Pittsburgh via the T. Unfortunately, my commute occurs during the middle of rush hour so there is no chance of me taking my bike. But here are some points to the discussion:
1. The 42S line is currently closed due to construction work on the bridge over Highway 51 (between Dawn and Palm Garden station). Thus, until September or so, the T only runs between Traymore station and Overbrook Junction. From Overbrook Junction, you have to transfer to the Willow station and take the train in. If you’re wanting to take the T in, you could ride down to the Willow station (located here)
2. Once the bridge is working again, the 42S line will run straight into the city again. As was suggested before, you might want to consider riding Broadway all the way down to the Fallowfield station (located here).
Fallowfield is a high-level platform to make it easier for you to load your bike on and off the T.
3. I would avoid Palm Garden as its a low level stop and not suitable for loading your bike. If you really want to go as far as you can, you can try to get to South Hills Junction station. It’s on the PAT busway which is a restricted road (located here). I’ve never ridden in that area so I don’t know how strict they might be; there are some sidewalks though so you could always walk your bike if needed to.