You can also use "top" in the console to see if there is a lot of activity. Also.. uhm... Govnah helped a lot to look at that, it draws graphs about CPU configured speed and system usage. But: Both drain a lot of battery themselves... so keep on a charger or so and see if you can see a lot of activity.
Also watching the log can give a lot of hints... for example corrupt contacts can send the contact linker into an endless loop of crashing and being restarted by activity manager. That will drain your device fast, too.
Your second question: This again hints in the direction that not your battery is at fault but something else in the system. Something keeps the system so active that it will always consume more than the 500mA power a USB port is allowed to deliver. If you have multiple devices on USB ports that are powered over USB or even on a USB-powered HUB, then there is even less power avaliable for the Pre3. Minumum power was something around 100mA, IIRC.
The Pre3 charger allows 1A = 1000mA power, the TP charger even allows 2A power and I'm pretty sure it charges a Pre3 faster, too (did not verify that 100%, but was my feeling).
So if the quota issue does not resolve the device-busy-issue, you have to keep on searching. Your battery probably is still fine.
@bethel95 & for power requirements and C+Dav: The version probably (not sure what caused your battery issue) does not play too much of a role for that. I did not have abnormal battery usage with 0.3.18. More important is, if the syncs are running smooth. Very bad would be a device that is still working on initial sync. But also if there is a bug and the service crashes a lot, it has to do more work on every sync to recover from the error condition, that will drain a lot faster. As a rule of thumb: The longer a background task (not only C+Dav) takes, the more power it will consume. A usual C+Dav sync should take only a couple of seconds (you can check that from the app). If it takes a lot more *every time*, something is probably wrong and people should send me a log.
@TJs11thPre: there even is 0.3.23, now... mostly non critical fixes. I promise to look into Preware distribution, again. If this was completely my software, I'd do it already... but this is in the name of webos-ports and webos internals so I should probably use their feeds and also alpha/beta/stable testing stages (which is a very good idea for C+Dav in its current state!). But I have to learn the build process for that, before..