You can use the PC without the CMOS battery, because it is used for keeping the info like boot order, time, what hardware the pc had, etc. If the PC CMOS battery is dead, you have to go into the BIOS reset the boot order, time, etc.
So if you try to clear the CMOS, even the battery failure will not cause the PC does not boot.
When you do clear the CMOS, you need unplug the power cord, then either remove the battery or use the button to clear the CMOS, after that connect the power, boot the PC. If you can't boot the pc, then you may try Bench it - Troubleshooting
http://www.tomshardware.com/faq/id-1753671/bench-troubleshooting.html
One more you may check the cup socket to see there is any bent pin or not, if you only removed the cooler, and also removed the cpu, but if you don't touch the cpu at all, you don't need to check, but it does not hurt to check it out.
If you use only one stick ram with cpu+ cooler+ using the onboard iGPU, and the pc doesn't boot, the problem may from either cpu or MB. If you have other compatible parts to test, you can easy to find out cpu or MB has or has not problem.