I don't know if a burned out headlamp would cause other light problems, but it needs to be replaced anyway so you've got nothing to lose by trying that.

As for the original battery, I'd get that replaced ASAP. I went with an AGM from O'Reilly auto parts and have been very pleased with it. I'm not sure if they have O'Reillys where you are, but their Super Start AGM is a Deka made by East Penn Mfg. They are pricey, but I felt the AGM quality was worth it....

Super Start 48AGM - Battery 2005 Chrysler Crossfire | O'Reilly Auto Parts