I am building a circuit with an Arduino Mega 2560 and a large amount of sensors estimated to draw about 1.5 - 2 amps. Since this exceeds the 1 amp limit of the onboard regulator, I am planning on connecting an external 5V regulated power supply directly to the 5V pin of the Arduino. Can the 5V pin, traces, and Atmega 2560 handle 1.5 - 2 amps?
No, not a change.
And 2A for just sensors! Are those sensors heaters?
And 1A for the onboard regulator, not a change in heaven it can handle 1A. If fed by 12V it can handle the Mega + a hand full of led's and it's done.
You can use a 5V supply. Just connect it to the 5V pin of the Arduino and connect the sensors directly to the supply as well. One note, if the Arduino is powered from an external 5V, DON'T connect the USB.
Devices only draw the current they need, you can connect a 1000A 5V supply to any 5V device and expect it to work as normal (unless its got a short circuit, in which case bad things happen).
You should route the 5V supply directly to the devices, as well as to the Arduino. The current shouldn't be
flowing through the Arduino at all. Same for ground connections.
What kind of sensors though? If high currents are involved there may be issues with IR losses in
the cabling adding small voltage errors to the sensor outputs (if analog).
I have 8 MQ gas sensors and a touchscreen LCD display. The MQ sensors have a heating element and the data sheets seem to put their peak heating power consumption at around 800 mW each. Would perfboard work for routing the 5V and GND connections to the sensors? I have only used breadboards before since I am pretty new but I'm pretty sure a breadboard can't handle much current.
Perfboard is designed for that 
But 800mW is still only 160mA and I don't know which touch screen but combined you're probably <300mA. I would indeed not draw that from the Arduino but it's an order of magnitude less then 2A 
septillion:
Perfboard is designed for thatBut 800mW is still only 160mA and I don't know which touch screen but combined you're probably <300mA. I would indeed not draw that from the Arduino but it's an order of magnitude less then 2A
Reread that post. He has 8 MQ sensors. I'm guessing with the touchscreen it's close to 1.5A, so a 2A supply is in line with what is needed.
Ah, yeah, I read it as "I have a 8MQ-sensor". 