Well, aside from these specific sentences you make a number of other English mistakes that a native speaker would not make. For example, the expression is "day to day" and not "day a day".
Otherwise ... You should understand that there are many different English dialects, each of which has its own preferred way to say certain things. None of these are wrong, just different, and an expression that might be common in one dialect might be completely absent from another. It all depends on who is talking.
All of your sentences are grammatically correct. None of these is more common than any other.
I prefer dancing to swimming
I prefer to dance rather than swim.
I prefer to dance rather than to swim.
Some are more wordy, but perhaps the best way to clearly say what you mean:
I prefer to dance the tango rather than to dance the mambo.
"Would prefer", grammatically, is no different from "prefer" in that you can use either the infinitive or the gerund:
I prefer dancing in the moonlight to doing my taxes.
I would prefer dancing in the moonlight to doing my taxes.
I prefer to dance in the moonlight rather than do my taxes.
I would prefer to dance in the moonlight rather than do my taxes.
Of course there are many other ways to express a preference, and some people prefer never to use the word "prefer" at all. So I think it's not very useful to say one expression is "less common" than another, but rather understand, from listening to native speakers, in what context each is used.
(Edit) To be clear the "to" in "I prefer to dance rather than (to) swim" is part of the infinitive "to dance" and "to swim" and not part of the expression "prefer to ..." My point is that you can use either the infinitive or the gerund ("-ing" form) -- both are used equally, although sometimes one sounds better than the other.