Two tips: 1) It sometimes helps to rephrase the sentence into an equivalent English-sentence that looks easier to analyze. 2) Often times, you can break down the sentence to make it easier to parse. If you have trouble wrapping your head around the sentence, try phrasing it in a slightly more suggestive way. For instance:
"Every grandparent is such that either they have only daughters, or they have exactly two sons, or they have no children."
In general, "Every $\varphi$ is such that $\psi$" gets translated into the predicate calculus as $\forall x (\varphi(x) \rightarrow \psi(x))$. Your $\varphi(x)$ here is "$x$ is a grandparent", whereas your $\psi(x)$ is "$x$ either has... (etc.)". So overall, the translation should look like this:
$\forall x(x \text{ is a grandparent} \rightarrow x \text{ either has only daughters, or exactly two sons, or is childless})$
So if you can figure out how to say "$x$ is a grandparent" and "$x$ either has only daughters, or has exactly two sons, or is childless", then you'll know how to translate the sentence.
How do you say "$x$ is a grandparent"? Basically, it amounts to saying that $x$ has some child, who also has some (other) child. So this just amounts to $\exists y (C(y,x) \wedge \exists z(C(z,y)))$. This formula (which has $x$ free btw) is your $\varphi(x)$, which goes in the antecedent of the conditional of your universally quantified sentence.
How do you say "$x$ either has only daughters, or exactly two sons, or is childless"? Well, it seems to be a disjunction about $x$, so split it up into cases: if you know the whole thing is a disjunction, you can tackle each disjunct separately and then put it all together with $\vee$s at the end. So you just need to analyze "$x$ has only daughters", "$x$ has exactly two sons", and "$x$ is childless". Hopefully, things are clear enough that you can do these on your own.
It looks like all three are trick questions, and the best answer to each of them might be "this meaning cannot be expressed in propositional logic".
Sentence (a) speaks about necessity. Your suggestion $\neg(p\to m)$ is logically equivalent to $p\land \neg m$, in other words "I will pass philosophy, and by the way I'm not taking notes". That is something quite different from saying that notes are not necessary for passing. Propositional logic cannot in itself speak about necessity -- I've gone on at length about that in an earlier answer.
In sentence (b) you have found the problem yourself -- the naked truth of the entire sentence doesn't at all depend on whether you want soup or not. The only slightly defensible propositional rendering would be simply $s$ itself, but that entirely fails to encode the real content of the sentence, namely "... and you're welcome to eat it". Propositional logic cannot express permission either.
Sentence (c) is just using "equivalent to" in a casual, decidedly non-logical way. In ordinary conversation, the meaning of this sentence is a value judgement, namely that the moral desirability of eating at McD is no higher than the moral desirability of destroying rainforests (not "rainbow forest", I think). Propositional logic is unable to express moral judgements or desirability.
It is possible that the point of the exercise is to let you discover for yourself some problems that modal logics attempt to address (especially if there's modal logic later in your course). Or it may simply be to make you aware of the dangers of translating natural language to logic by thoughtless pattern-matching.
Best Answer
I can understand that the use of 'cannot' is a bit confusing ... it seems to be stronger than just saying that David and Emily are not both happy: they may not both be happy now, sure, but to say that they cannot both be happy seems to say that they can't ever both be happy, i.e. that it is impossible for both to be happy.
In fact, in modal logic you can express these kinds of stronger claims, where:
$\square P$ means "It is necessary that P is true"
$\Diamond P$ means "It is possible that P is true"
Using those symbols, translating "David and Emily cannot both be happy" can be done as:
$\neg \Diamond (r \land p)$
or, equivalently:
$\square \neg (r \land p)$
But, I assume you are currently not doing any model logic at all, since you are just starting with propositional logic. As such, you should really just treat the sentence as "David and Emily are not both happy"
Good for you for noticing that those two sentences are not quite the same thing though!!