I'm beginner in Japanese, and I was analysing some kanji. I was noticing the parts of 燃. I see a dog there, a fire, another fire, and then there is this 夕 with two strokes in the middle instead of one, that apparently, as far as I could look it up, doesn't actually have a meaning, or does it? If not, is there many kanji with meaningless parts like this? How does it come to be?
Good question, with an unexpectedly complex answer.
'夕 with two strokes' is indeed a radical you can look up -- in particular it's the 肉 (meat) radical. Radicals can have different appearances (variants), just like the two appearances of the fire radical 火 in 燃.
燃 is a radical-phonetic compound made up of the fire radical 火 and the right hand side 然. In turn, 然 is composed of the fire radical as well as the components 肉 (meat) and 犬 (dog).
The reason why the fire radical appears here twice is this: the meaning of 然 in Old Chinese really was the same as 燃 (to burn)! Later in the development of the Chinese writing system, the sound of 然 was borrowed for its present meaning, and then the original character had the fire radical added to represent its original meaning.
A similar process, where for a certain character a new meaning comes to replace the old meaning, and a radical is added to represent the old meaning, happened to many other characters. One example is 雲, the original form of which was 云 (which is now used in its borrowed meaning of 'to say').
The original meaning of 然 was "to roast dog meat over a fire", which was broadened semantically to "to burn". Hopefully it's now clear why there's a 犬 in there.
To answer your general question though: "do all kanji parts have a meaning", the answer is no. A character can't necessarily be broken down entirely into semantic "atoms". See the Wikipedia article on phono-semantic compounds for an example of non-semantic character components.
Yes, not all, but many parts constituting a 漢字 have meaning(s).
If you wish to learn what an individual element constituting 漢字 signifies, I recommend you read 「常用字解」 written by Dr. Shirakawa Shizuka, published by Heibonsha, as a handy and instructive manual.
常用字解 explains the origin of 燃 as follows:
The base of 燃 is 然, which signifies the action of burning (roasting) the meat of dog being offered as a sacrifice to the Heavenly God. The 月, a part of the character is a simplified form of 肉 (meat).
However, 然 grew to mean 然れども (but, however) side by side to the original meaning of 「然」 later, and it became necessary for Chinese to differentiate the usage of 然 from 'however' to "burn' by developing a new character. So, they added 火 (fire) to the existing 然 as a quick method. The creation of 「燃」 was made in Hang Dynasty after the 2nd century B.C.
An additional note in response to some of the comments and questions below:
月 when used as a 偏 （the left-hand）and 脚 (bellow part of 漢字) instead of 月 on its own, it’s called 肉月（にくづき), meaning the letter of 月 representing a part of body in such a way as 腕 (arm), 肌 (skin), 肺 (lungs), 腹 (abdomen), 胎（womb), 肋 (libs), 腰 (waist)，股 (crotch), 腿 (thigh), 腸 (bowels), 肝 / 胆(liver), 背（back), 胃 (stomach), and 腎（kidney）.
Good question. Actually I thought about that myself several times and could not come up with a clear answer. Here I will just try to answer to your specific case.
If you look up 燃 on some dictionary, such as here, you will find that it just reports the radicals (部首) that in this case are 火 and 灬 (both fire).
Anyway, if you look 燃 up on a different dictionary such as Midori you will see that it actually you get a link to 然 and interestingly it decomposes this the character in 月、犬、and 灬. And if you look closely it makes sense. The top left part could indeed be a moon (EDIT: apparently it is actually "meat" and not moon, they just look the same).
Edit: So at least for this kanji yes, each part has a meaning apparently. As pointed out in another answer it seems that Midori's interpretation is not even correct though.
I could not find an online version of Midori so I will put a screenshot here: