Yup, you're on the right track here!

I'd do it much the same way. Start by rephrasing the question as

"Is (a+b)/ab > 1/(a+b)?"

S1 tells us that (a+b) is positive, so we can rephrase the question as

"Is (a+b)²/ab > 1?"

If ab is positive, then the question is really

"Is (a+b)² > ab?"
or
"Is a² + 2ab + b² > ab?"
or
"Is a² + ab + b² > 0?"
or
"Is a² + b² > -ab?"

Since ab is positive, -ab is negative. a² + b² is never negative, so in this case the answer is YES, (a+b)² > ab, implying that 1/a + 1/b > 1/(a+b).

The second case is if ab is negative. For this to happen, exactly one of the two variables must be negative. But then |a| + |b| could not equal a + b, which we know is true, so this scenario is impossible.

S2 just says that a > b, but it doesn't tell us anything about the positivity or negativity of each number, so we can't do much with it: as you say, it leads to two different questions but provides no information with which to answer those questions.

If you ever get stuck on one of these conceptually on the test, I'd just pick some numbers to illustrate likely scenarios (both positive, both negative, one pos one neg, etc.) and see if they all give the same result. If they do, the statement is likely sufficient; if they don't, the statement is definitely insufficient. You won't be certain of your answer, but you'll improve your odds of being right and buy time for other hard questions to come.