I am the subject of the verb, but English treats me as an object. You can be used for both.
So OP's "reversal" principle doesn't really mean anything. If I say "I like you", I can't reverse it to *"You like I" any more than I can say *"Me like you" (unless I'm Tarzan talking to Jane).
In "If I were you", obviously "I" is the "subject" and "you" are (is?!) the "object". If you reverse the roles of the pronouns "I" becomes the object, so it has to be "If you were me"
See Are You and I You and Me? for further exploration of the issue.
The first form "If you were to go home, you would feel better." should be grammatically correct, but it sounds rather strange to me.
The second form "If you went home, you would feel better." is grammatically absolutely correct and also expresses the right thing. It is a so-called Conditional Clause of Type II which means that the event in question (i.e. you go home) is improbable but still possible. In general such a clause is constructed according to the pattern: If + simple past, would/could/might + infinitive.
The third form "If you will go home you will feel better" is incorrect. If you slightly adjust it to "If you go home, you will feel better." you get a so-called Conditional Clause of Type I which expresses that the event in question is likely to happen. In general a Type I If-clause follows the pattern: If + simple present, will-future or can/must/might+infinitive or imperative.
There is also a Type III, which, in your case, would be "If you had gone home, you would have felt better." It implies that the event in question is impossible, because you are talking about the past. In general, Type III follows the pattern: If + past perfect, would/could/might + have + past participle.
Other conditional if-clauses that do not fall into one of the above categories are usually grammatically incorrect. As always, there might be some exceptions and special cases, but the above is definitely a good guideline.
EDIT: People also sometimes speak of a Type 0 if-clause which addresses something that is generally true, for example: If it rains, I take out my umbrella. The construction is fairly simple, as you see.
Best Answer
"Are" is a present-tense form of the verb to be. So saying "If you are a bird..." is an observation about what a person is now:
Although nonsensical, it is following the logic that you and the other person are the same, so if the other person is something, you must be the same.
"Were" is a past-tense form of the verb to be, however, it is also used to hypothesize about something that might happen. This usage is normally recognisable by the conditional "if".
You might say:
This is a purely whimsical statement - saying that if you were a bird, hypothetically you would fly away. There is no logic here.
This doesn't really work with your example statement, because it cannot be "logical":
Your hypothetically being a bird does not dictate that someone else would be one too. I think for this example you need "If you are....".
Your other example though is not about logic:
This is entirely hypothetical because it is about what you would do if another condition was met (the other person was a bird). The only thing wrong with this example is that, when speaking about a hypothetical future, "will" should be "would":