Yes, "that sucks" is extremely informal, usually used by younger people, and the previous person who answered is right--it takes great cultural sensitivity to know how and when to use it, and with whom! When you learn a language, I feel like it is important to know what slang means, but I don't advise on using it yourself, unless you have a very very high level of the language. A safer response would be "Oh no." or "Wow, that's too bad." I would personally respond, "Oh no, are you feeling better?/How are you doing?"

Yes. In fact, I think you should. "That's bad" is plain English, can be used at any time in any situation, and can communicate any degree of emotion with context and tone of voice. "That sucks" is informal English, and you need cultural sensitivity to know when it is the right expression to use. It's not usually offensive, but "that's bad" is much safer.