Yes - they are backward compatible - but the max speed of the card will be lowered to the PCI-e 2.0 standard (not a problem for most cards since nothing really saturates the 2.0 spec currently)
Incorrect, there is no speed difference between 2.0 and 2.1. PCI-e 2.1 adds the new management features for PCI-e which will be used mostly in the 3.0 cards (whenever that standard is released) but there is no difference in speed between the two types. There is however a difference in speed between 1.0 and 2.0 and a medium to high end graphics card used on PCI-e 1.0 will encounter a throughput bottleneck. To my knowledge no graphics card exists which requires a 2.1 slot and will not function on a 2.0 slot.