Not sure I follow what you mean by the battery indicator flashing five times. It only flashes one LED when the battery is low and not charged! You should press it again and see eight little LED's light up in a line. When you plug in the MagSafe cord they also will light up (maybe only a few if not fully charged).
If the MagSafe charger LED is green then you have a charging logic issue within your logic board as SMC it telling the charger the battery is charged.
How about giving us a better view here on whats happening, download this great app: CoconutBattery and paste a screenshot of the main window so we can see what its' telling us Adding images to an existing question
Did you get your system wet at some point, something spilled into the keyboard or trackpad? Or, sit your system down on a wet surface?
Well as you already know the battery condition is poor (4.4%)
It's so bad it makes sense just taking it out for now. That way it doesn't interfere with your MagSafe usage. Ideally, you really need to replace it. The blinking battery tester LED's is telling you the battery is beyond hope. Your menu battery info panel is telling you the battery needs to be replaced and the blinking battery checker LED's is likewise telling you the battery needs replacement.
Diagnosing your other issues are a lot easier when the low hanging problems are removed.