The answers are right, and so is the information you read about positive and negative meanings. The key here is the word 'semantically' - the question of whether it's a good or bad thing in the real world is irrelevant. The word 'few' is negative semantically in that it means NOT MANY. Obviously, having 'few friends' ( not many friends) is a bad thing, while having 'few problems' (not many problems) is a good thing. The phrase 'a few' is positive semantically in that it means SOME ( i.e. not zero). So, while it is good to have 'a few friends' (as opposed to no friends), it is bad to have 'a few problems' (as opposed to no problems). I hope that makes sense.

1. A few. 2. Few. "Little" and "few" (without "a") mean "not much"/"not many". As a rule, they are negative ideas. She has few friends here (not many, not as many as she would like). "A little" and "a few" have a more positive meaning. I enjoy my life here. I have a few friends. But in your sentences the words "unfortunately" and "lucky" are key words. Unfortunately, she has got some/a few problems. She is lucky, she has not got many problems/has got few problems. "Few" and "a few" are used with plural nouns. "Little" and "a little" are used with uncountable nouns.

yes, what you have checked is right. 1. A few, 2. Few. A few means several or more. Few means only a small amount. if you put 'few' in 1. it would change the meaning of the sentence to say she only had a small amount of problems, where as a few problems makes it sound like she had more than that, a lot more.

Unfortunately, she has a few problems <-- "got"? So lucky. She has few problems. <-- "got"? . a few = stresses a number greater than one few = stresses not many . Why? Grammar describes what is. The accepted use is as explained above and by others below. On eneeds a convention to convey meaning. This is the convention here. The collection of such understandings is called grammar.