Поехать is perfective verb, i.e. it has no present tense at all. Поехали refers to a completed action, which is the start of moving.
Also, do any other verbs of motion prefixes do this, or just the prefix по?
Sure, there are quite a few. But being perfective, they also refer to some completed part of "moving action". For example, отъехать: Поезд уже довольно далеко отъехал от станции. This means that the train went quite far from the station, i.e. it has finished some part of the distance, but it says nothing about whether it stopped or was still moving.
Thus prefixes with motion verbs show which part of moving was finished. If you mean the journey was fully completed, you have to use prefix при-, like in приехать, прийти, прибыть, прилететь, приплыть and so on.
Yet note that sometimes поехать still could serve for the same purpose as simply ехать. That's because the speaker silently implies that after "they have started moving" immediately follows "they were moving", which is simply omitted but should be obvious from the context.
UPD. Now on the matter of "Сейчас мы поехали в квартиру". There's the obvious "time contradiction" between "сейчас" and "поехали", but this is quite tolerable in Russian. Though it makes the sentence ambiguous: it has (at least) three different meanings depending of context and pronounce.
- Determination Now we shall go to the apartment. Here past tense is to show speaker's resoluteness (he thinks the future to be as definite as the past);
- Narration Then we went to the apartment. So simply, "сейчас" is for "narration present time", "поехали" is "regular past tense" (as @Quassnoi mentioned, this could also be seen as a remnant of old "proper perfect" tense);
- Small interval Last time we went to the apartment [but what about next time?!]. Сейчас (or now for that matter) is not "a point" but rather "a small time interval". So it's possible to match it over the past or future tense (like Now I will sleep), if speaker believes they are close to each other.