1) Nonsense: Say there are two laws, one which states that the first level of denotation is a member of a higher order type or a class, than the objects which are in or referenced by it. The second says that an element must belong to a distinct set, and cannot belong to the subsets of that set. When these laws are broken you create paradoxes and nonsense. Sense thus presupposes nonsense as types of expression for which rules against infinite regression and disjuctive syntheses are based, as they can kill sense. When the description of the two series of "word = x", "thing = x" explains the functioning of denotation, manifestation and signification, it's eliding the explicit description of the rules that prevent sense from not being created. This chapter is an explanation of that which is later explored explicitly with the chapter on paradoxes, but in the form of the raw material of expression that doesn't follow the rules that produce sense. In a way these are multiple presentations of the same process, just from different angles. The ultimate point here is to uncover the process by which sense is produced, and to explore in what sense sense operates on the surface of things.
2) The phantasm is a bit more difficult to explain in simple terms but I'll try..
Try to recall the first time you were to exposed to sexuality --something that left a clear mark on your development psychologically. That scene as emotionally experienced by a subject leaves some latent as well as explicit mark on ones psyche would be considered an example of a phantasm.. Something that lingers, that operates on multiples levels of your mind (conscious and unconscious). Another example might be some early experience where you're acting out some type of aggression or urge for freedom from ones parents and you're marked by an originary experience of fighting or doing something risky that leaves resonates intensively inside your body, and marks your psyche without being the slightest bit conscious of it... and that comes to signal at some unconscious way who you end up becoming. That originary marking is the meeting of your biology with individual expression of culture (human psycho-social creation). Those are phantasms. You continually create/repeat. You are caught up in them.
These are essential to understanding the messy depths of the latter half of the Logic of Sense but you've already passed the chapters on language, orality, sexuality, you're in the heart of the pathic as the bodily conditions for the arrival of thought.