That indicates a problem with the hardware of your system. Many computers will do this if you do not have a mouse and keyboard attached. It will usually display a message in these situations - something along the line of "Keyboard Error - Press <F1> to continue. The system clock misfunctioning can/will certainly cause this which is why they replaced the battery on the motherboard. Replaced properly with new battery, it should last for several years, so it probably isn't the battery at this point although someone could certainly test it to make sure it is good. If the battery is good and all peripherals are properly connected and functioning then it would appear that your motherboard is failing and will need to be replaced.
The machine turning off during a game is most likely an indication that something inside the computer is over-heating, most likely the CPU or the graphics card, although it could be a component on the motherboard or even faulty RAM. You should verify that all of the fans in the computer are working properly. You should also remove any dust and debris which have built up inside the computer. You can use canned air to do this. You cold also use a low powered vacuum to collect the dust, however, do not let it touch any of the circuits inside the computer. The tinest static discharge and damage something on a motherboard, rendering the enitre machine useless.