Hi,
"Bistrot" is the room and have to be writed with a final "t" because you can write "Bistrotier" (the owner of the bistrot). But anyway the french people accept two forms, also "Bistro" and "Bistrot" are used. The definition of Karim look to be the best ethymologie.
"de Paris" is right, and the explanation with a vowel is absolutly ok. But in this case you can use a short form of "de" then "d" if you want.
I would write: "Le bistro(t) de Paris" (look well) or "Le bistro d´Paris" (sound better for a smal local)
Friendly,
TJ