The drop of the /ə/, is as usual in Quebec as it is in France (although it does not happen in the same places). It is more common with the 1st person singular pronoun “je”. It may be devoiced before a verb with a voiceless consonant initial. This is most notable in verbs normally beginning with an [s], as the well-known example je suis ‘I am’ that is often realized as “j’suis” or “chu” ([ʃy]), or je sais ‘I know’, realized as “j’sais” or “ché” ([ʃe]). Since the drop of /ə/ is not exclusive to Quebec, this phenomenon is also seen in other dialects.

It just makes the sentence shorter/faster to say. it is used in familiar contexts. In formal contexts, you try to pronounce the whole word. But from experience (in Quebec), I can say that many people still drop the e when in formal contexts.

Now for when you can drop a mute e between two pronounced consonants, there is no real rule. You’ll have to listen to people talk to know when you can and can’t drop the e. It’s like when you say c-ya instead of see you or I’m gonna instead of I’m going. A non-english speaker would’nt know about it unless he heard someone say it like that.

Is this something that “just happens” when you’re speaking conversationally?

Yes.

When is the mute not dropped between two pronounced consonants?

Less casual contexts. When you want to insist. If you are slowing for any reason (because you are thinking about what you are going to say). If one ask you to repeat.

This is not French per se, since this phenomenon of disappearing so-called mute e tends to happen a lot less below the Loire river. The accent of the Southern France is famous for keeping almost all the e, it has even been the subject of stand up comedy shows from prominent performers from the region of Marseille (see there for instance http://www.youtube.com/watch?v=m7h1HE6KPO4).