The variance caused by traffic lights is generally higher for me than the variance caused by my own speed. How many lights are there? Are there alternate routes that can help?

The good news about a route about lots of traffic lights/traffic is that the bike time can be really competitive with the drive time. The bad news is that there's only so far you can improve without running into the delays caused by the lights.

You'll probably have to give us more stats on the hills to tell us what's "nominal", which varies wildly. In general, though--yes, you can go faster.

I find that my speed in the "city" (more suburban, really) averages about 10-11 mph. This includes traffic lights, everything door to door. A bit higher if there's less climbing or lights, probably a bit lower with more climbing. My 5.5 mi AM commute has about 500 feet climbing (most of it in about a mile, cumulative), and I average about 30 min. Plenty of lights on that route, though it's no city center with lights every block. My ride home is about 25 min, so the elevation has some effect.

I don't think, personally, a computer will help much. I had one for a while, but it's not like I can't figure out that I need to pedal faster when I'm slacking. I do use a GPS on my phone for kicks, but I don't look at it while riding. Just noting your departure/arrival time is probably just fine. You just need to push harder and go faster.