Can I run Intel Core i3 on a DDR2 motherboard?
No, your old motherboard can't accept newer Core i3/i5/i7 processors.
Will ddr2 graphic card work on ddr3 motherboard?
I was stuck on something like this ages ago it is the connection between the two you need to worry about one is gaming memory one is physical memory they are separate and don't care about each other. What you want to look at is your motherboards pci is it like 2.0 etc. if this matches you gpu it should be fine.
Can my DDR2 motherboard support a DDR3 or DDR5 graphics card ?
Yes it should work. No problem. Only thing to check before getting that GPU: Does your motherboard use an AGP slot or a PCI-E ? Most recent GPU's use PCI-E slots, and thats the only thing that would prevent it to work, if it just simply does not fit.