Macbook pro 2012 Auto power up when plugged in (Dead battery)

Question

My MBP 2012 automatically power up when plugged in. (I havn't changed any settings regarding the power and I have double checked it and confirmed)
My battery went dead before 3 weeks for no reason and stopped charging at 21%, but still the battery condition is displyed as normal.
Do I have to replace the battery with a new one and will it solve the problems. (Auto power up and charging issue) ???
PS: The warrenty is over and I don't have apple care either.
please check the infomation bellow:-
Battery Information:
  Model Information:
  Serial Number:          W02XXXXXXX
  Manufacturer:          SMP
  Device Name:          bq20z451
  Pack Lot Code:          0
  PCB Lot Code:          0
  Firmware Version:          201
  Hardware Revision:          000a
  Cell Revision:          165
  Charge Information:
  Charge Remaining (mAh):          1006
  Fully Charged:          No
  Charging:          No
  Full Charge Capacity (mAh):          4718
  Health Information:
  Cycle Count:          175
  Condition:          Normal
  Battery Installed:          Yes
  Amperage (mA):          0
  Voltage (mV):          11263
System Power Settings:
  AC Power:
  System Sleep Timer (Minutes):          0
  Disk Sleep Timer (Minutes):          0
  Display Sleep Timer (Minutes):          0
  Wake on AC Change:          No
  Wake on Clamshell Open:          Yes
  Wake on LAN:          Yes
  AutoPowerOff Delay:          14400
  AutoPowerOff Enabled:          1
  Current Power Source:          Yes
  Display Sleep Uses Dim:          Yes
  PrioritizeNetworkReachabilityOverSleep:          0
  Standby Delay:          4200
  Standby Enabled:          0
  Battery Power:
  System Sleep Timer (Minutes):          10
  Disk Sleep Timer (Minutes):          10
  Display Sleep Timer (Minutes):          2
  Wake on AC Change:          No
  Wake on Clamshell Open:          Yes
  AutoPowerOff Delay:          14400
  AutoPowerOff Enabled:          1
  Display Sleep Uses Dim:          Yes
  Reduce Brightness:          Yes
  Standby Delay:          4200
  Standby Enabled:          0
Hardware Configuration:
  UPS Installed:          No
AC Charger Information:
  Connected:          Yes
  Charging:          No

Answer 1

first I have tried SMC reset and PRAM reset.
Unfoatunatly now I can't do them bcoz the computer powers up automatically when I plugged in.
PS: I have tried another adopted and still it's the same