ddr memory speeds (Mhz's) and mainboards memory slot speed capasity

Hi!

I would need some information on the topic of ddr memory speed and how they are compatible with different mainboards.

For example, if the memory card is of speed 400Mhz, and the mainboard
specs. tell it supports Dual channel DDR2 800/667/533, will the 400 Mhz
memory card work on the mainboard, at the speed of 400Mhz, or at all?

Or if the memory card is 800MHz DDR2, and the mainboard only supports
memory cards of speed 400MHz, will the 800MHz memory card work on the
mainboard at the speed of 400MHz, or at all?

Also, if you have a 800/667/533 mainboard, and you install one 800 memory and another 667 memory, will the memories work good, with
separate speeds?