Recently our laptop (4yo) shut down unexpectedly after being used for 45min to 1 hour! I was wondering what the issue is? Is there anything which can be done by us? We are out of warranty.

If AC adapter is connected and the system gets shutdown automatically it is for sure that this problem is not for battery. You can try going to properties of My computer - Advance system setting - startup and recovery settings - automatically restart

CHECK THE FANS! The biggest enemy to your laptop is HEAT! You just replaced the display it's very possible you inadvertently disconnected the cooling fan. Clean it while you have it apart dirt and dust prevent proper cooling

First - re-seat the hard disk and reboot the computer. If the computer does not boot up then reboot and go into the BIOS configuration to see if the BIOS detects the hard drive. If it doesn't detect the hard drive then it could be faulty.
Then the hard drive needs to be replaced in the computer/laptop, Windows installed and the computer or laptop's device drivers plus your programs etc. to get a functioning computer/laptop again.

try using a cooler pad for your laptop for heating issue,
try changing the RAM also sometimes the faulty RAM also does this

When the CPU overheats then it will shut down (i.e. thermaloverload) and won't boot up again until it has cooled down. If this is theproblem is not fixed then the CPU will eventually fail.
A faulty fan causes the problem or the fan and heat sink assembly are cloggedwith dust.
Check the CPU fan to see if it spins freely, if not replace the fan.
If the fan and heat sink is clogged with dust the remove the dust and make surethe vents are clear and dust free. Use a small brush and can of compressed airto do this.

I had the same problem. Check your cable from power/volume board to motherboard. One end of my cable connector was damaged.

If the Computer restarts randomly, then the issue might be due to any damaged Operating System Files.
But, if the Computer shuts down randomly, it is due to the Processor.
All the Processors will be designed in such a way that, if there is any overheating issues, the Computer will shut down randomly.
The overheating might occur due to the malfunction of the Cooling Fan in the Computer.
As this is a overheating issue, it would be better to get the Unit examined at the Service Center.
Replacing the Cooling Fan in the Computer should resolve the issue.

Enter the BIOS when you first turn on your laptop, usually done by pressing F2, and make sure it is holding onto the correct time.

If it is not that, then you are probably looking at a heat-related issue. If you often work with the laptop sitting on a flat surface like a desk, you should really consider getting something to lift it off of the desk an inch or so. Or just buying an active chill pad that would run a fan underneath it.

Possible heat issue. Check the cooling fans and see if theyre running. Usually one on the bottom, back or side of the laptop. Make sure that the cooling vents arent blocked during use.

1) For sounds, you can check the sound properties and see if the balance is set right for the left and the right speaker. Alternately, move the balance to the left and see if you hear anything 2) There can be lots of reasons why the laptop would shutdown automatically. Does it shutdown or would just turn off without a proper shutdown. It can be due to some device - especially the processor getting heated or some device is not getting enough power for the system to remain up. Please revert back with further observations.