The first one is the most confusing one, because I have heard people say it
usually when they are upset. The other two examples to me sound a little
softer with だろう/でしょう. Therefore, I suppose the purpose of だろう is to make the
sentences, including the first one, sound a little less direct? The second
example is also interesting, because of the んだろう part. Would it be a big
difference, were the sentence written with only だろう, or even without だろう?

I read about the origin of だろう/でしょう, but it was not much of a help. I think I
might be missing the nuances of these words now, but might develop a feel for
them later on. They don't always translate well as "might" or "maybe" or "I
wonder" etc.
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nuances
What is the purpose of だろう in expressions such as 何をしてんだろう or なんてかわいい犬なんだろう or 何でだろう
だろう at the end of a sentence containing interrogatives ("wh-" words like 誰, 何,
なんで, etc) means "I wonder". While an ordinary question is addressed to others,
だろう(か) forms a question addressed to yourself. でしょう is the polite equivalent.
Please remember the basic patterns below:

  * 彼は来る **だろう** 。  
I believe he will come.

  * 彼は来る(の) **だろうか** 。  
I wonder if he will come.

  * **誰** が来るの **だろう** 。  
**誰** が来るの **だろうか** 。  
I wonder who will come.

  * 彼は来るのか。/彼は来るのですか。  
Will he come? (direct quesition)

So 何をしてんだろう ([contraction of
何をしているのだろう](https://japanese.stackexchange.com/a/61959/5010)) means "I wonder
what they are doing". It _could_ be translated into English as "What are they
doing?", especially when the speaker is upset, but it's important to remember
this is not an ordinary question that directly expects an answer from someone.
Likewise, 何でだろう is "I wonder why". Unlike 何でなの (which directly asks "Why?"), a
listener doesn't have to reply.

なんてかわいい犬なんだろう is "What a cute dog!". Although this だろう is still a "I wonder"
marker here, this is a special _exclamatory_ construction you have to master
separately. This pattern always starts with なんて, なんと or なんという. See: [JLPT N1
Grammar なんという / なんと / なんて](https://jlptsensei.com/learn-japanese-
grammar/%E3%81%AA%E3%82%93%E3%81%A8%E3%81%84%E3%81%86-%E3%81%AA%E3%82%93%E3%81%A8-%E3%81%AA%E3%82%93%E3%81%A6-nan-
to-iu-nanto-nante-meaning/)

Related: [だろう in question
context?](https://japanese.stackexchange.com/q/62238/5010)
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