Can I catch Pokémon when I move very quickly?

Yes this is possible, after tapping the Pokémon, your movement speed does not affect the catching period.

I have been catching them during hi-speed train travels myself without any problems so far.

update

Since the last update, Pokémon spawn rates during high speed travel have been reduced significantly. This to prevent people from playing and driving. The limit seems to be somewhere between 60 and 70 Km/h. During travels between 15 Km/h and 60 Km/h a reduced catch rate is observed as well, in combination with a higher escape rate.

TL;DR: It is still possible to catch Pokémon while moving fast, up till certain speeds, and with reduced efficiency.

Yes, if you tap the pokemon it doesn't matter if you are a long distance away. I do it myself during public transport.

I noticed a huge decrease in catch rate. I used an incense on the highway traveling 100kmh. I encountered 28 pokemon and only caught 8. Most ran away after the first poke ball.