I know of key makers who can clone remotes and cut any type of key but we are talking of your everyday remote fob with basic functions but even for those they need the original in order to clone. In this case the fob is lost and even if he did have it, it is way too sophisticated for the local places here to replicate.
Why the transmission was taken out was perhaps some ustaad advising OP that the jerks (misfires) were caused by a bad tranny. Only OP can tell us. If Dewan had a go at diagnosing, it might have been coils. Who knows. But my gut says it would have been a bad head gasket. These cars are known for head gasket failures.