Your first example is grammatical, the second and the third aren't.
"I would take the dogs out for a walk means that you may perform this action in a future moment if the opportunity arises, if you have the time to do it, and so on.
"I would have taken the dogs out for a walk indicates an action which did not take place in the past (for reasons which are not explained).
"I would took the dogs out for a walk is not a possibility in English, as you cannot use a simple past after "would".
In a different context, another possible meaning for the first sentence is something that you did frequently in the past, an action which you happened to perform and which was discontinued later on. Perhaps this is what you meant by your third definition.
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
If I went home for dinner, I’d have a soft drink is an example of the Second Conditional (I’ve changed the wording slightly to make the English sound more natural). The Second Conditional is used for a situation when the action envisaged in the if clause is unlikely. It is normally used to say what might happen now or in the fairly near future. However, the construction could also be used by a speaker describing something in the past, particularly in some kind of narrative. Used this way, it suggests that on those occasions on which the speaker went home for dinner in the past he was in the habit of having a soft drink. If I went home for dinner, I had a soft drink can be used in the same way (but it cannot be used as an alternative to the normal use of the Second Conditional).