Are Ukrainian "швидко," German "schnell," and English "swift" cognates? Even if they're not, it could be a useful mnemonic.
I think they might well be :) I would rather compare it to the German "Geschwindigkeit" rather "schnell"
"Geschwindligkeit" is a noun meaning 'speed' where as "schnell" is an adverb.
Closer to home, I wonder whether it is cognate with the Polish szybko. It must be, but it seems to have drifted further than other words here like rzadko рідко często часто
Colloquial Ukrainian even has шибко/шибкий itself with the same meaning, but you don't hear it very often
When must I use іти and when їхати with vehicles? Example, can I say "Цей поїзд іде швидко" without changing the meaning?
I would say it is a more colloquial way. We do use іти for vehicles sometimes..
Is «іти» used more when talking about routes ("This train goes from Ashford to Basingstoke") while їхати is used in other cases?
I heard that from some forum users (then it is similar to their use in Russian) but I am not sure if it applies.
It does apply.
What not "quickly"?
The same problem
Can we use "drive" for the train?
The train itself doesn't drive, the person drives it. So no :)