No.
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They should be able to trade.
yes
yes, but not to any of the old games, only between other silver versions, gold versions, or crystal versions.
Trading between an DS and a DS lite is the same as between a DS and a DS. There is no difference.
No, you cannot trade between the two different versions because the Pokemon are different or the Pokemon know different moves that are not in red version. When they re-make gold and silver for DS there may be some way to trade between Pokemon fire red and leafgreen to the new gold and silver but I doubt it.