What is the purpose of past future tense?

First of all, you cannot have only one 'flag' (as you say) turned on, there should be at least two, and here is why. Continuous and Perfect do not really define the timing, they can happen in present, past and future. Continuous denotes a started and not finished, in progress action. Perfect, on the other hand, describes something the result of which we can observe, so that action is already complete.

Now let's get to your examples:

• I would go: it is easy to understand this if for a moment we go back in the past and consider it to be present. For this example, we will have I will go. Now let's add some more context and move it back to past: I thought I would go with him, but he insisted on going alone.
• I would be going: again, bringing it to present: When you get home I will be going to school. That means that at some point I will be in the middle of doing something (going to school). And now moving the sentence to past: I told you I would be going to school.
• I would have gone: if I say it now, it will be If you are a little late I will have gone, the difference between this and the previous case is that in this case the result is important not the process of the action itself, and the previous case stresses on incomplete progress of the action. Again, moving it to past: If you were a little late I would have gone.
• I would have been going: here it is both perfect and continuous, so both the result and the progress are emphasized. Coming to present will give By the time you get married I will have been going to school for 5 years. Now moving back: I always thought I would have been going to school for 5 years by the time you graduate.

Thus, I would go means that at some point in the past you could say/think "I will go". That is actually another good usage of all the future in the past mystery - indirect speech. When you convert direct speech to indirect, you need to bring the speech to appropriate timeline, for example:

• Tom said: "I will never go there again".

  converts to

• Tom said that he would never go there again.

I would have gone, as I already said, means at some point in the past you could say/think "I will have gone", which in its turn means that at some point in the future you will be gone, the fact that you won't be there any more is the most important statement.

Would has another meaning also, not sure how that is called in grammar, but it is used to express some usual, habitual action, similar to used to:

• I used to eat an apple every morning. == I would eat an apple every morning.

Regarding

My english teacher says that past tense are often used to say something that's not true or regret. Really? How?

It's the subjunctive mood you are looking for. It uses a tense earlier than that in a true condition to indicate things unrealized.

1. I am 30 years old and have several kids.

   • Then I say in subjunctive mood:

     If I were 20 and without kids, I would travel around the world.

   • Compare this to the truth:

     I am 30 years old and I can't travel around the world.

2. I lost my lottery and it turned out to be a winner.

   • Then I say in subjunctive mood:

     If I hadn't lost my lottery, I would have made a fortune.

   • Compare this to the truth:

     I lost my lottery and I didn't make a fortune.