Will a 12 VDC 1A Power Supply Adapter supply too much power and damage an Arduino Mega 2560?
No. You have everything wrong there.

First off, a power supply does not provide power. It provides voltage. The device attached to it then draws current. As much current as it needs at any particular moment in time.  The current rating of a power supply is the maximum that the power supply can give.

The same goes for Arduino IO and power pins: The current ratings are the maximum current you can draw from the pin.

Exactly what current it will draw from the power supply depends completely on what is connected to what pins, and even what calculations the Arduino is doing at the time.  To get a ball-park figure of current requirements you need to know the current requirements of whatever is attached. If a device has its own power input that is the current you need to worry about. If a device is powered directly from an IO pin (such as an LED) then that forms part of the Arduino's current (which starts at about 40mA or so for the board itself).

There are many other factors to take into consideration:


The 5V regulator on the Arduino can't give more than 1A maximum, and considerably less than that when powered from a high voltage like 12V.
The DC jack on the Arduino is limited to 1A due to a polarity protection diode.


It is usually recommended to use a separate 3.3V regulator for an ESP8266 module (or 5V if your module has a built-in 3.3V regulator) due to the lack of power capacity in the Arduino's on-board regulators.  

As a rough guestimate:


Arduino: 40-60mA
ESP8266: 200mA (when transmitting)
PIR: Negligible
Relay: 50mA when active


So you'd be looking at 300mA as a minimum. More is better. 1A should be more than sufficient. 2A would be overkill but give a big margin of error. 

For everything you have there, it would be better to use a 5V power supply, not a 12V power supply.