Is there a term for music composed of everyday sounds (animal calls, cars honking, etc.) that are timed just right to sound like music together? (I hope this isn't off-topic here)
This is very often called musique concrète, or avant-garde as a broader term. The technique you are mentioning is also referred to as ready-made or found object (objet trouvé in French). A ready-made is an object/sound/whatever that already existed without any artistic context, which is then turned into art by giving it that artistic concept.
Two relatively well-known examples are Revolution 9 by the Beatles, and Several Species of Small Furry Animals Gathered Together in a Cave and Grooving with a Pict by Pink Floyd (yes, that is the title of a song).
In Revolution 9, you'll notice that the sounds aren't actually aligned to a musical rhythm per se. This is usually the case with musique concrète: you throw all rules about music overboard, and create what you think sounds good.
In Several Species (...), the sounds are not actually natural; they're all created by Roger Waters tapping/slapping his own body in various places, making weird sounds with his voice, and speeding all of this up quite drastically. In this song, everything is aligned rhythmically, so it's a good example of how far you can go with this kind of sound: you could even adapt the pitch of each sound to create whatever melody you like. Anything is possible with this kind of stuff, as long as you're imaginative and creative enough. :)
You can call it in various ways, depending on the style of production / performance.
It can be called Found Object if you are using sounds like you mentioned without performing it.
It can be called Experimental, if you are improvising it - sampling it live.
You can call it Experimental Electronic if you are using your computer / digital sampler.
You can call it ambient,
While not a term for this genre as a whole, this thread would be incomplete unless someone mentioned the practice of sampling.
Hip-hop and electronic musicians (as well as those in other genres) often sample sounds from other music, lines from TV shows and viral videos, and environmental sounds like rain and car horns. Sometimes the sounds are autotuned and/or edited to have a certain melody, and other times they are left as-is and the interest comes from how the new song recontextualizes the sampled content. Wikipedia has a good article.
Sampling has spawned many debates about authorship and originality, and the way that hip-hop artists borrow samples from each other (often without giving explicit credit) has been used as a metaphor for the internet and open-source practices as a whole. See remix culture and Cory Doctorow, Information Doesn't Want to Be Free.