(1) is an ordinary expression used when asking something to others.
It is common sense to use polite expressions when asking for something to others. It seems that polite words/expressions such as "お願い" for "願い" and "します" for "する" are used in (1).
"します" for "する" is certainly a polite form, but concerning "お願い", which I'll explain later, can not be said so in (1).
We don't use (2), but use (3).
(3) is used when the person who is higher in position or status like the company's superior asking something to the person like his subordinate.
As for 願いする in (2), it is not a Japanese expression. As you know する sometimes makes a noun a verb like お願いする（お願い+する）, but 願い cannot become a verb by being added する. Therefore the expression with (2) is not used or is not a Japanese sentence because 願いする doesn't make sense.
I'll provide more information on the difference between 願い and お願.
Regarding the basic form of a sentence using "願い" is like "願いが かなう/実現｛じつげん｝する/成就｛じょうじゅ｝する wishes come true". Furthermore, when the basic form is modified, it'll become like "「私」の「日本語が上手になりたいという」願いが 実現する/成就する/かなう My wish that I want to be good at Japanese language comes true".
On the other hand, regarding "お願い", though "よろしくお願いします" is often used, it is not a complete sentence. "「Aさん」に「Ｂ」をよろしくお願いします" is a complete sentence composition. In general, Aさん is often "you", so omitting "you", there are many cases that we use "Ｂをよろしくお願いします". Furthermore, as B is often clear in the conversation as a context, "Bを" is omitted and the expression with "よろしくお願いします" is commonly used.
Therefore, simply the phrase with "よろしくお願いします" means "「私たちがいままで話してきたこと what we have been talking about」 を 「あなた」によろしくお願いします".
You can use 願い and お願い at the same time to create the following sentence.
However, it is incorrect to say "私は神様に私のお願いが実現するように願いする"