There are lots of second person pronouns in Japanese, and of course, there are occasions where you are supposed to use them. I think the reason your teacher advised you to avoid using them at the beginning is not because you are not supposed to use them but because it will be difficult for a beginner to pick up the right one. Rather than making a wild guess and picking the wrong one, do not use them for the time being until you proceed to an advanced level and become sure which one to use in which occasion. That was your teacher's message, I guess.
あなた has a very subtle nuance, and can only be used between certain relations, contrary to what another answer says. If you use
あなた to your teacher, it would be definitely rude. But that does not mean that
あなた is an impolite form. It actually is an honorific form, and why it becomes impolite has a complicated reason behind it. It is true that you often cannot find a pronoun with the appropriate politeness, and have to call the second person by the name or by the title. As I mentioned above, the rule behind this is very complicated, and you probably will not get it at this point. So I think it is a good idea to follow your teacher to avoid them, but that does not mean that they are not used in real Japanese. They are just too difficult for beginners to use.
Your question: "[D]o Japanese people in Japan actually pay attention to these distinctions?" shows that you are optimistic about how much Japanese language is sensitive to social relations. Japanese is one of the languages in the world that reflects the social relation in various aspects of the language most heavily. Yes, they do. More than the speakers of most other languages.
To show you how severe it is in Japanese to use the correct expression to refer to a person, I will point out that, the distinction that you mention for the second person is just the beginning, and when it gets to the third person, it becomes even more complicated. You can either refer to a person by the name like
山田, which is not (particularly) polite, or put a polite affix or a title after it like
山田先生, but which to use depends not only on the relation between you and the person referred to, but also on who you are talking to. Japanese adopts a system called relative honorification (as opposed to absolute honorification used, for example, in Korean). If
山田 is your boss with the title
部長 within a company, and you are talking with someone within the company, you have to be polite enough to refer to him/her as
(山田)部長 (even attaching the polite affix like
山田さん is not polite enough, and is rude). But when you are talking with some outsider, then you have to say
山田, and saying
山田さん would be impolite to the outsider (and the expression
山田 does not become impolite to