p.s. would a grammar then be necessary if we all spoke telepathically?
Sure. If you have two robots that can communicate wirelessly and send information directly to each other, there must be some format to that information. That format can be considered a grammar. There would likely be different properties of that grammar, but you'd still need structure.
However, you might mean do we need "linguistic grammar" and perhaps not. In that case, we might just need thought, which has its own structure. (Although some would say thought and language are the same at some level, but that's controversial!)
To start with, I guess, could you please explain the interrelations between a grammar, its PF representations, and its LF representations? It seems to me that LF representations are the "ends" of all language. We use grammar and PF "means" to reach that end. So LF representations depend on PF representations which depend on a grammar. Is that basically right?
What you're referring to has been developed over a long time with slightly varying terminology and smaller or larger changes along with changes in theory. At least one name for it is the "T-model". Just searching for that I found this paper which seems useful:
http://cogprints.org/5432/It's presenting something else as an argument, but the background information on the T-model is useful. It's a bit older, though, so it's not exactly what is current in the MP now. But the MP is built on all of that, so unfortunately to get what's going on in the MP you need to understand the history. (Again, the MP is an attempt to streamline the older theories, so you have to start with them.)
Personally I've discussed and learned more about that level of syntax from the perspective of [Formal] Semantics (dealing with LF). So that's another area to explore for information.
I do think that your average textbook introducing the MP will explain at least the basics of this model though.
Note that one major complication is whether there's order to it or not. Are there merely endpoints, or is the lexicon "first" then PF and LF "last"? Taking "generative" literally, it seems to be directional. But often that's just a metaphor. Read specific authors carefully to see.
Also, I'm not exactly sure what the principle of full interpretation or what converging at LF or PF means.
http://www.glottopedia.org/index.php/Full_InterpretationAs I understand it, anything that ends up in PF must be represented in the pronunciation, and anything at LF must be included in the meaning. PF is the interface with pronunciation (or articulation more generally) and LF is the interface with semantics.
Also, could you please give one example of a "crash"? I appreciate your help!
Crashes occur when the derivation can't continue or when something goes wrong. There's nothing wrong with lining up words in a certain way (or in MP terms "merging them") until you find that features don't check or there's a violation of some principle. An example depends on having a specific theory in mind, but the idea is straightforward. Let's say that an element requires an accusative complement (so the verb "eat" wants some object in the accusative). If its (intended) object is in the nominative, the derivation will crash. One way to explain that would be to say that features can't be checked. Another would be to make some sort of principle that objects must be in the accusative.
As I said, unfortunately to really understand the MP you have to understand the history. Almost nothing of what you've asked about is unique in the MP, and in fact it operates exactly the same way (but in a somewhat different context) in earlier approaches like GB.