07 Starting Issues

[alfabob]

Hello, wondering if anyone is having similar starting issues. 2007 yaris sedan with under 40k, just installed new battery a few weeks ago. When starting the car, there are a quick series of clicks until the engine begins to slowly crank. After about 3 - 5 cranks it starts up and idles around 1500rpm until driven around for ~15 mins, then drops to the usual 500-700. What is confusing me is that if the alternator was bad, the battery should have been dead by now (have driven it around at least 10 times with head lights and accessories on). Initially thought it was due to cold weather, but the issue persists even when temps are in the 40-50's. Any ideas?