1.
Let A = John will wear a shirt
Let B = [He'll] be cold
Let C = The sun will shine
(A&B) V C
~(A&B)
-----------
C

The third sentence is a conclusion, so it needs to be distinguished.

This would work: (A&B) V C ; ~(A&B) ∴ C


2. AB is short for A&B, it's taken from college algebra, where one omits the multiply sign: a * b = ab

For #1, it is usually necessary to break up paragraphs into separate arguments. They serve as the premises for your conclusion, the function of the entire proof that you want to evaluate.

In your example, I translated it like this.

(J & C) v S
~ (W v C)___
S

The problem you gave was a classic disjunctive syllogism. However, the second line (i.e. ~ (P v Q) ) can be either ~(WvC) or ~W&~C because if anything it is an extension of the DeMorgan replacement rule.

As to your second statement, in propositional logic if a syntactical composition is missing a connective, it usually means that it is not a WFF (well formed formula). This is a definite rule in propositional logic that as far as I am aware of is non debatable.