How can profile (log the charging and discharging current for some time period) my battery?
This is not as in-depth as FireLord's answer, but I found a few apps that might be useful.  

These all appear monitor current and have a log feature:  


Current Widget: Battery Monitor by RmDroider   
Battery Utility Free by SUXD   
Advanced Battery Monitor by Codeka   
Battery Monitor Widget by 3c   
Battery Spy by Vertiform Technologies


These are more general system monitors but they appear to include battery stats and logging:


Elixir 2 by Tamás Barta   
AndroSensor by Fiv Asim