Thinking too much into it

To address your main question, which is a KEY concept to understand for data sufficiency:

If we are given preliminary information in the question stem, then we take that information as immutably true. We never question it.

So, when a question reads "If 4r + 2s =12 ...", we read that as "for the purpose of solving this question, 4r + 2s = 12 MUST BE TRUE".

Any information in statements (1) and (2) gets treated the same way. Our task is to determine, based on the known truth of the information in the statements, whether we can definitively answer the question.

Looking at the specifics of this question, Cramya has hit on the key concept to understand: the "n distinct linear equations for n variables" rule.

If you've read some of my previous DS posts, I'm sure you've heard me talk about this rule before. I love this rule. If it were legal to do so (I've heard that Vermont is considering it), I'd marry this rule. This rule is the key to DS bliss. In fact, every time you see a DS value judgment question (as opposed to a yes/no question), you should think about how you can use this rule to solve.

Let's review the Question stem:

If 4r + 2s =12, what is the value of s?

We see we have 1 equation and 2 unknowns. We think "if we get 1 more distinct linear equation that doesn't introduce any new unknowns, we can solve".

(1) r + s = 5

Yay! One more equation, let's run through our "safety checklist":

a) does it introduce any new unknowns? NO
b) does it have any exponents or other wackiness? NO
c) is it just the original equation in disguise? NO

Sufficient.

(2) 2r + s =6

Yay! One more equation, let's run through our "safety checklist":

a) does it introduce any new unknowns? NO
b) does it have any exponents or other wackiness? NO
c) is it just the original equation in disguise? YES!!

If we multiply both sides of this equation by 2:

2(2r + s) = 2(6)
4r + 2s = 12

So, statement (2) doesn't give us any new information at all: insufficient.

(1) is sufficient, (2) isn't: choose (A).