"Do they have a bicycle?"
Translation:У них есть велосипед?
Some languages (Russian included) show possession by saying that X is at/by/of/on Y to mean "Y owns/possesses/has X".
Here, у is a preposition that generally indicates proximity (so it can mean "with", "at", "on", "upon" etc.), них means "them" (prepositional form of the word "they"), and есть means "there is." Thus, this sentence could literally be translated "Upon them there is a bicycle", which just means "They have a bicycle."
Edit: A better explanation (from what I previously wrote) is because "у них" is a prepositional phrase: the bike is "upon them," so to speak. "Their" is an adjective, strictly speaking, and possessive adjectives are not used in Russian, at least not in the same way they are in English.
I think that's a better reply than "That's just how Russian does it."
When bicycles were first invented "velocipede" was the generic term (including in English) for vehicles propelled by pedalling. It derives from the Latin for "speed" + Latin for "foot". As design settled on the 2-wheeled version (bicycle as opposed to the unicycle or the tricycle) English vernacular picked on specifying the wheel arrangement as the normal way of referring to your velocipede.
In short: yes, it directly relates to "velocity", from the same Latin word.