This is going to be long, so bear with me.
One of the main reasons the passive voice is used in language is to be able to control 1) the flow of information and 2) what the subject/topic is at any given moment. Stephen Pinker gave a great explanation of this in English in one of his lectures which you can find here.
Basically, certain sentences or parts of those sentences just sound weird or abrupt unless the subject is the person or thing being acted upon.
1) Control of information flow
Take the English sentence:
I picked up the baby, who was then stung by a bee.
The first part of this sentence is in the active voice, I picked up the baby.
The second part, however is in the passive voice, the baby was stung by a bee.
If we banned all use of the passive voice, this sentence would become something like:
I picked up the baby, and a bee stung him.
or
I picked up the baby, after which a bee stung him.
In the first example sentence, the flow of information goes
"Me → Baby → Bee"
But in the second set of sentences, it goes
"Me → Baby → Bee → Baby"
Not only does the focus of our attention change more frequently, two of those changes are to the exact same piece of information; the baby!
This is why the passive voice sounds more literary to our ears; it provides us with a smoother flow of information.
2) Active control of the topic
The passive voice also lets us control who or what the subject of a sentence is:
Take the sentence:
After the search party found and rescued my brother when he got lost on a hiking trip, he went on to found a charity that helps rescued climbers pay for their rescue services.
The subject of the sentence starts out being "search party" in the first half and changes to "my brother" in the second half.
What if it's not necessarily important who found him? What if we want to focus entirely on the brother?
In that case, we could say:
After my brother was rescued when he got lost on a hiking trip, he went on to found a charity that helps rescued climbers pay for their rescue services.
The subject of the entire sentence is "my brother", which we accomplish by putting the first half in the passive voice.
The example sentences you provide show use of both.
I'll go through them one at a time:
友達の赤ちゃんを抱っこしたら、泣かれてしまった
Instead of this, we could say:
友達の赤ちゃんを抱っこしたら、((その)赤ちゃんが)泣いてしまった
In the case of your example sentence, the flow of information is smooth:
(I’ve used an overscore to denote the speaker; an underscore the baby)
（[私が]{HH}）[友達の赤ちゃんを抱っこしたら]{LLLLLLLLHHHHHH}、（[その赤ちゃんに]{HHHHHHH}）[泣かれてしまった]{HHHHHHHH}。
In the case of the second sentence, the flow of information is less smooth:
（[私が]{HH}）[友達の赤ちゃんを抱っこしたら]{LLLLLLLLHHHHHH}、（[その赤ちゃんが]{LLLLLLL}）[泣いてしまった]{LLLLLLL}。
For your second example sentence, you have:
雨に降られて、服が濡れてしまいました。
If we were to write this without passive voice, we would say:
雨が降ったため/ので/降って、服が濡れてしまいました。
In this new sentence, we first hear that it's raining, and only then do we obtain the information that the something (服) got wet as a result.
However, in your example sentence, we first get the information that something is getting rained on. We don't know what that something is, but it is nonetheless the subject of the sentence, which is then almost immediately revealed to be 服. Not only does the subject stay the same throughout the sentence (active control of the subject) and give it better flow, but in this case, we engage our listener by making them wonder what the subject is before we give it to them.
For your third example sentence, you have:
父に死なれて、大学を続けられなくなりました。
If we were to remove the passive voice from this we would get:
父が死んだら/死んだ後、大学を続けられなくなりました。
In the first sentence, the subject remains on the speaker for the entire sentence. By doing this the speaker's father didn't just die, we as the listener are forced to empathize that the father dying was something that happened to the speaker, something that affected the speaker. This is why this use of the passive is sometimes called the passive of suffering as @Aeon Akechi mentioned, because it forces us to empathize with the suffering of the speaker.
