Battery service and insanely low maximum charge
You need a new battery.

This is confirmed with the following two conditions as stated in the question:



  When checking it further, I found that my battery status is "Service
  Battery"


  I used coconutBattery to check and found that the Full Charge Capacity
  of my Mac is insanely low



Your battery, according to it's manufacture date, is over 3 years old, so it's not out of the norm for it to begin failing at this time.

Replacement batteries are fairly inexpensive and fairly simple to replace ("simple" as in not a complex procedure), but it's considered "difficult" because it's glued into position so it requires diligence to remove.  ifixit.com has an excellent step by step guide if you want to tackle this yourself.