It's not the motherboard battery. It could be the motherboard. But I'm getting ahead of myself.
The symptoms you're describing indicate the system is probably not completing the POST (power-on self-test). There are countless reasons this could be happening. Here's the basic troubleshooting I go through in such a case, and they are an effort to narrow down where the problem resides.
Shut down the PC, disconnect all the cables and pop the case open. Stick your head in the case and smell. If you get a very acrid scent in your nose -- which can only be described as 'that burning electronics smell that sticks with you for days and you never forget' -- things are very bad. Otherwise, blow any and all dust out of the case, preferably outside.
At this point you could try connecting the PC again to see if the problems are fixed. You would be surprised how often simply cleaning the dust from a PC fixes serious issues. It's also a good time to check to be sure that all the fans are spinning when the power is on. If only the power supply fan is running, that gives you an idea of where the problem is. Assuming that doesn't help...
Taking note of where things connect, disconnect the CD drive and hard drive from the motherboard and power supply. Remove any adapter cards. Disconnect any USB headers for the case. If you have onboard video, even remove any discrete video adapters and connect your monitor to the onboard video. Otherwise if the discrete video card is the only video adapter in the system, you must leave it in place.
Remove all but one stick of RAM, but make sure the lowest numbered slot is still populated. That is, look for writing next to slot. On some systems, it's the closest to the CPU or back of the case, and on others it's the furthest from the CPU or back of the case. Look for slot
0. Leave RAM in slot
0 and remove the rest.
Examine the system board. Look for any obvious signs of burning or damage. Look at the capacitors (the cylinders) and see if any are bulging or oozing liquid.
Connect the power and monitor to your computer and turn it on. What we have done is taken your computer to the bare minimum required to boot a computer: CPU, RAM, display, motherboard, power. A computer requires these things (and only these things) to complete the power-on self test. There is no input device so after the BIOS show the computer will probably complain about no disk, having nothing to do, or display a blinking cursor in the upper left corner of the screen.
If the computer completes the POST in this case, turn off the computer and reconnect something. I'd start with the remaining RAM, if there was any. Keep going until you find the device that breaks the computer. Either that device is faulty or the device it connects to is partly faulty (for example, the disk controller on the motherboard might broken, but it only causes a problem when a disk is connected to SATA-0). Replace whatever you find is causing the problem. In our example, I would replace the disk and the cable connecting the disk first, then the motherboard if I discovered a known-good disk caused the system to fail.
On the other hand, if the symptoms persist on the computer when all you have connected are the basic components components, then the problem is in one of those components. At that point, there are some rules of thumb.
RAM is the most common component of the basic system to fail. A computer with four sticks of RAM has four times the likelihood of a RAM failure. Motherboards and display adapters are in the middle. CPUs almost never fail except when something else does (the motherboard, power supply, or the CPU fan).
When in doubt, replace the cheaper part first. Not just because it's easier on your pocket. It's also because more expensive components tend to have more rigorous QA testing.
Computers are built like buildings. An apparent problem on the top floor can actually be caused in the basement. The opposite is almost never true. The basement is the power supply. First floor is the motherboard. The motherboard is the true core of a computer, as that is what everything connects to and intercommunicates with. Second floor is the CPU, RAM, and video adapter. Third is the rest of the hardware like disks, adapter cards, displays, keyboards and mice. Fourth comes operating systems and networks. Fifth is application software. At the very top is the user. You'll notice the higher the floor, the more common the problems are and the less severe they tend to be in terms of system usability (all jokes about users aside). You can also see now why it is unwise to buy a low-end motherboard and power supply when you're getting high-end processors and video cards.
When diagnosing a faulty system, start at the bottom floor and work your way up. Often it's easy to jump to the fourth floor, but occasionally errors there are not caused there.