Since you are on windows 7, try this: Go to device manager find something called " Microsoft ACPI Compliant Control Method Battery ", reinstall it, restart and give it another shot.
Assuming you're using windows, Go to Control Panel then System & Security then Power Options then on your left Change when computer/monitor sleeps. Adjust it to your preference.
Are you certain your replacement battery is compatible with your laptop?
As for capacity, it really depends on the tasks you're running on your device. It's fairly old and I'm guessing even modern day browsing may be straining it.
The specs you posted above are anything but cost-effective. A better way to choose a laptop is to decide what price range you can afford or you are willing to pay up to. In any case, you can find a list with compatible Red Hat laptops here.
There's a good chance that you're experiencing thermal throttling. Check the temperatures during gameplay with something like msi afterburner. If the fps drop when the gpu & cpu reach certain temperatures, you'll know for sure.