Learn English – “Can have” vs. “could have”

You can safely omit it pretty much always.

I think it's largely a regional variety issue. As the article explains, in the UK "I've got" might be used more commonly where Americans would say "I have".

I remember being taught phrases like this in elementary English classes (due to British English emphasis in Finnish schools at that time):

I've got a cat.

But rest assured, it is never wrong to omit "got" in a phrase like this and just say:

I have a cat.

Also in those cases where it's used to add emphasis or indicate obligation...

I've got to go now!

... you can always omit it and still be perfectly understood (you can add the emphasis in other ways):

I have to go now!

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.