savestates aren’t compatible across cores and may not be compatible across even the same core after an update, depending on what’s been changed.
snes9x is the most broadly appropriate core. bsnes-balanced offers a few accuracy improvements and a few (terrible) games work on it that don’t work on snes9x. The -mercury cores are slightly faster than the regular bsnes mainline cores and they offer the ability to use HLE special chip emulation instead of LLE, which can be very important in maintaining full speed on some hardware (for example, the cx4 LLE in Megaman X2/3 is particularly demanding for very little apparent gain). Snes9x 2010 has some speedhacks that make it a little faster than mainline snes9x but that’s probably not going to be necessary for your machine.