Strange Starting Problems

I purchased this 2010 Hyundai Accent back in May of this year. Been alright, just a lil daily driver for a delivery job, good gas mileage, whatever. Have had a series of problems in the last couple months, and especially the last few weeks.
>July, Battery/charging system light comes on and off randomly. Get an electrical system test at shop, everything passes with flying colors. Ok.
>September- That light is now basically always on. Get tested again, still passes test perfectly.
>Now - A few hitches of not wanting to start right away. Get tested again, same results. Suddenly, the engine sometimes stalls when idling, though not often, maybe once a day. Get tested again, same results.
Now, it's getting more difficult to start, though it turns over every single time- what seems to work is pumping the gas pedal while turning over, after a few pumps it finally fires up. Electrical system still checked out perfectly fine a few days ago. Is my fuel pump dying, or am I just losing pressure? And the time interval between startups seemed to only be after sitting for a while at first, but now it seems to just do it if left for longer than a half hour.