I notice English must not be your first language so excuse me if I misunderstand what you are asking here...
What do you mean by random touches? I am taking this to mean when a certain screen is open random touch events happen on screen as if you are tapping it yet you are not.
I am not sure where Game Mode comes in and what exactly you mean.
Replacing the screen may not have fully fixed your issue. Keep in mind there is both your screen and the digitizer (the sensor that activates when you actually touch the screen) and you also have highlight lamps. Anyways replacing these with after market stuff is always a crap shoot. You could have gotten a lower grade screen (LCD get rated A B C D by how many dead pixels or other small issues are allowed to be present in a screen before sale). You may have even gotten a screen not truly made for your device. These can all cause different issues in different ways. I have seen heat be a factor (where if your CPU temp is high the screen is warm and this changes how/when touches get detected). I mean you could have even accidentally left a tiny piece of glass in there and it is randomly pushing on the screen, or the data cable when you connected it if it is not seated all the way or if the contacts were shorted out from getting wet this again could cause odd random issues.
As far as adjusting touch sensitivity that is a device setting any user can get to:
ANdroid has a feature that lets you control the sensitivity of your screen. While it's not an accessibility feature per say, it could be useful for people with dexterity issues. If you want to be very sure you are touching the right thing, set the screen sensitivity lower and you will have to work harder to activate things. If you need a softer touch because you just can't muscle your screen around, set the sensitivity higher. When I started seeing lollypop devices, I noticed that they really took some serious muscle to get them going and now I have begun doing this to all of the devices I see so that they work better with talkback.
To change this setting:
Tap Language and Input.
Scroll to the very bottom of these settings and tap pointer speed.
I have seen sevreal default speeds, none over %50. Increase the slider to make the touch screen more sensitive and easier to tab. Decrease the slider to make your touch screen less sensitive and more difficult to tap.
Tap OK and then experiment with the results.
Samsung Galaxy Note3 Lollypop,
Motorola moto G lollypop,
Motorola Moto X Lollypop,"
The place where you find this setting also depends on which version of Android you are currently using so if you do not see it you can google Adjust touch sensitivity Sony Xperia
I did find this specifically for the Xperia, but if you have the Z2 or if you have a different OS version than what this users used you may again have to do some more searching.
"You have simply follow some of the steps after that your problem will be solved:::::::::::::::>>>>>>>>>>>
First of all go to the main menu
Then go to the setting icon
Then go to the display option
Then tap on the touch option
Then increase the level of touch
Then save the setting
After that restart your device
All the best