Hey Evan! Good to see you!

I understand the opening passage a bit differently:
In the Paris of small neighborhoods,
Forgotten by tourists,
Postcards,
And their old snapshots,
The Eiffel Tower plays a trick on the eyes of passers-by.
You'll note I've taken the opportunity to punctuate this bit because it desperately needed it. In fact, the whole thing needs it, in both French and English.

l'boulevard Voltaire > It can remain a boulevard in English.

Qu'aux Champs Élysées > "On" here, too. They're both streets.

Faire de l'oeil > This is an idiom. It usually means, "making eyes at" somebody, like flirting. It's kind of a cute way of putting it in this context.

Belleville's market > I think this should be "Belleville Market."

If you look here, it points you off to here, which leads me to believe the bit about the "hammam" needs to be translated as "Turkish bath."

flâner > I think this translates more often as "wander about."

Loin des clichés
Des cartes postales et des vacanciers > I know "clichés" can also mean the same word in English, but given the references right below that to, once again, postcards & tourists . . . with their inevitable, ubiquitous cameras . . . I think the other meaning, "snapshots," is the one intended here. And I can see good arguments for having a comma after "clichés" as well as not having one.

qui me font tant vibrer > I would go with "really make me come alive" here.

Marché des Enfants Rouges market > I don't know. I think if you're going to keep "Marché," you can dispense with "market." On the other hand, I'd be tempted to do it the other way around, dropping "Marché" and keeping "market." And I might be further tempted to make it "Market."

musics > I know it's plural in French but this word practically never occurs in the plural in English. I might go with something like "various styles of music."

In the Paris I like I'm going about > In this Paris, I like to wander around.
The dictionary gives "balader" and "flâner" as synonyms.

This is a connected passage:
balader
Au gré des jardins,
Des portes cochères, et des escaliers.
Jeez I wish this had punctuation.
She's saying that she likes to wander around at the whim of what she accidentally happens to see, which determines where her aimless steps will take her. You'll note that "portes cochères" first of all is misspelled in the lyrics (similarly, it's "escaliers" in the plural), and second it's a unit, not two independent words. I've found several references that should explain what exactly it is:
http://www.wordreference.com/fren/porte%20coch%C3%A8re
http://forum.wordreference.com/showthread.php?t=84304
http://forum.wordreference.com/showthread.php?t=1869681
http://forum.wordreference.com/showthread.php?t=2079899
http://fr.wikipedia.org/wiki/Porte_coch%C3%A8re
http://en.wikipedia.org/wiki/Porte-coch%C3%A8re
Just to make things super-interesting, the term exists in English, but it doesn't mean the same thing as in French, so that means you do have to translate the expression and not just leave the words as they are. I would go with "double doors." That seems to be the consensus in the discussions. They are no longer necessarily used to allow the passage of horse carriages or cars, but the big double door is, evidently, a common occurrence in the architecture in Paris.