Adding a battery to my computer's PSU
A car battery is a single 12V source. But your computer's MoBo requires 12V, -12V, 5V, and 3.3V. So it's not possible to connect a backup car battery directly parallel to a PC PSU's 12V output.
Also, a car battery needs ~14VDC for being charged. So a PC PSU's 12V output cannot charge the battery.