I can see that. I spend most of my time on the top of the brake hoods. It seems most comfortable when I have the bars rotated "up". However, that causes the compromise that when I am in the drops, there isn't much bar to grab onto and reaching for the shifters is a long reach.

Bar end shifters? Mind if we ask about your bike? What make, sizet, etc. And you (height, etc.)?

Other questions: 1) When the width of the bars is listed (e.g. 44cm), is this usually outside - outside or center - center? 2) When they list drop & reach, how is this measured?

Center - center. Just about everything on a bike is measured center to center, unless specified otherwise.

Do people have preferred brands or models? I think that I would like to stick with carbon fiber.

I recommend you stick with the bars you have unless you need to change a vital dimension like 'reach'. There are only three kinds of bars: classic, ergo and compact. All of them have the end of the bar level with the heel of the bar at the top. If you want the end of the bar to project beyond the heel of the bar like in my photo you have to rotate the bar. Good carbon bars are not cheap, cheap carbon bars are not cheap. My bars are not carbon. But the compact bend is becoming nearly universal.

Our current bars are very ovalized just outboard of the stem clamp. I assume this is to spread out the load on one's hands. However, I spend very little time in this position so this feature isn't important to me.

Those are called "wing" models and the shape is supposed to help you when climbing.

Here. More than you wanted to know about handlebars: Tech: How to Choose the Correct Road Bike Handlebar Size & Shape - Bikerumor