Answer number 2 is correct.

Here is a link that explains the usage of "rather than" and "instead of." It seems that one should use the gerund (-ing form of the verb) if one cannot form a parallel structure with the first part of the sentence. In order to use a form other than the gerund in your example, you would have to use the infinitive without "to": "give," which is not one of your options. https://www.englishrules.com/writing/2005/rather-than-and-instead-of/

The answer is 2. Note that gerund phrases such as "giving full details of all their problems" always act as a nouns; for instance, in the sentence, "Giving full details about your problems is so annoying.", the gerund phrase "giving full details about your problems" is the noun and the "thing" / "act" / "process" which is annoying. Since "I'm fine, thanks" and "Could be worse" are also noun phrases, I believe that the parallel structure of this sentence isn't technically violated here. :) See this article about gerund phrases and how they always act as nouns. http://www.chompchomp.com/terms/gerundphrase.htm

#2 is correct :)