Daffys version: https://www.dailymotion.com/video/x1ylhor
Starts at 00:30
Well now! That song takes on a whole different meaning when Ella sings it. Especially towards the end there when Pop is shown to maybe not be all on the up and up.
Is it me or what Daffy sings, Merry-Go-Round Broke Down and Woo Woo all sound juuust a little bit different from each other?
Bugs has a theme song of sorts - I think it was used only once:
What’s up, Doc? What’s cookin’?
What’s up, Doc? Oh, you’re lookin’
For Bugs Bunny bunting
Elmer’s gone a-hunting
Just to get a rabbit skin…
Whoops! The rabbit’s gone again!
What’s up, Doc? What’s cookin’?
Hey! Look out!..Stop!
You’re gonna hurt someone
With that old shot-gun!
Ehhhh… What’s up, Doc?
My favorite Elmer’s gone a-hunting bit is when Daffy does the can-can to
*"So this is gay Par-ee,
come on along with me,
we’re stepping out to see
the Latin Quarter!
Put on your French beret,
we’ll sing the Marsellaise,
and guzzle wine just like it’s water!"*
Not sure about Bugs’ song but Daffy’s is definitely the Merry-Go-Round melody. There’s also Eddie’s bit from Who Framed Roger Rabbit.
So, the Looney Tunes theme comes from a song about hooking up at an amusement park.
I would not have called that.
I use the Looney Tunes theme as my phone’s ring tone. The Pink Panther theme announces my pill-time alarms.
(Mostly) To the tune of the William Tell Overture:
*Over hill and over dale
We’re always on the dusty trail
Hunting fox and hunting quail
Heigh ho! I’m a hunting fool
Giddy up giddy up giddy up my horse and now you’re up the final creek
Giddy up giddy up giddy up just like the wind I ride my forward steed
Sure of foot sure of eye peeling onions makes me cry, this makes no sense so do I…
So don’t you go and beat me daddy to the nearest bar, yeah!*
-Yankee Doodle Daffy, 1943
*If you’re on a highway and Road Runner goes beep beep.
Just step aside or you might end up in a heap.
Road Runner, Road Runner runs down the road all day.
Even the coyote can’t make him change his ways.
Road Runner, the coyote’s after you.
Road Runner, if he catches you you’re through.
Road Runner, the coyote’s after you.
Road Runner, if he catches you you’re through.
That coyote is really a crazy clown,
When will he learn that he never can mow him down?
Poor little Road Runner never bothers anyone,
Just runnin’ down the road’s his idea of having fun.*
The other Warner Brothers theme is Merrily We Roll Along, which they mostly used for Merrie Melodies cartoons. I don’t remember the song being used in the body of any cartoons the way they did with The Merry-Go-Round Broke Down. The chorus begins:
You can hear it here (over and over). The only recording I could find with lyrics was by Eddie Cantor (who co-wrote the song), but it’s so noisy that it’s annoying to listen to. You can see the lyrics and play the sheet music here.
Next up, Merrily We Roll Along…
Oh, cool. I never thought that I would get a chance to show off the following bit of obscure trivia! 
That song is called “The Latin Quarter.” It was written by Harry Warren and Al Dubin for the 1938 Warner Brothers movie Gold Diggers in Paris. Would probably be completely forgotten today, except for its being re-used in several WB cartoons.
Here it is as originally shown in that movie. The familiar lyrics begin at about 3:15 (after a big Busby Berkeley dance number!).
This cartoon can be viewed here. It’s a Porky, not an Elmer.
The lyrics are a little different, but I prefer my version. 
As a child, and to this day, I heard ‘cut the lights’. That is, dim the house lights.
ETA: Yep. Still sounds like ‘cut’.
.
I wonder how many kids out there in TV land heard that wrong? Half? More? I wonder what the original writers and singers think about that?
I wonder if the lyric is officially ‘curtains, lights’, but they changed it to ‘cut the lights’ for the actual performance. Because it definitely sounds like the latter to me, and I played that part several times.
I would assume the reference is to the stage lights, which would not be “cut” at the beginning of a show.