Normally the Hard drive can multitask alongside music and graphics but you'd need a fast processor or DMA-based hard drive controller to do it with.
As for games having this problem, I'd say are they originally intended to run from a floppy? If so, that might explain things. Some floppy games originally had copy protection and getting them to install to a hard drive takes some nasty magic tricks to get it to work.
To center the screen is a hardware-based option. Most games bang the hardware indescriminately so there is no way to adjust the way they access the video hardware. If you have a fast Amiga and plenty of memory you might try the classic Amiga version of
E-UAE to run some of your floppy-based games. This will let you simulate an A500 better and run from a workbench window on a graphics card. However it is generally slow to run games on an emulator.
-edit- According to the specifications in your signature neither of your systems is up to running EUAE. My suggestion is you get an '030 accelerator card and some memory for that A1200 since you have an A500 for the old floppy-based hardware banging stuff.
