[I think the city did that to cars, actually.]
Since this is in past tense, shouldn't think be thought?

[I nodded and wiped some spilled gas up with a rag and looked at him, “Are you from the city Sir?”]
As the part before the comma is not a speech tag, the comma should be a period.

[I asked, started some small talk of my own.]
"starting," not "started."

[“Seeing family. I should’ve figured,” I remarked, probably louder than I should have remarked, “Do you want your tires changed Sir?”]
As the latter dialogue is not a continuation of the previous sentence (they are separate), the comma after remarked should be a period, and even if it had been continuing the first quote, "Do" wouldn't be capitalized.

[“Sure Boy, why not?]
Is the capitalization of "Boy" intentional? That's the narrator's name?

[“Sure Boy, why not? Don’t got a long ways to drive. But I need to get back, you know?” he said jokingly with that ever-present smile, “And why should you have figured was going to see some family?”]
Again, the second piece of dialogue is a sentence of its own, so the comma after smile should be a period.

[I don’t think most people do, “Well I don’t really know Sir.]
Thinking most people do isn't a speech tag, and therefore the comma should be a period. Also, introductory words, like "well" in this case, require a comma after; and, terms of address require a comma before them, so a comma is necessary before Sir - and was the capitalization of Sir intentional, like the capitalization of "boy?"

[That’s when I recognized what made this man different than the other city men, I think, “I suppose the city does have its allure.]
The part before the dialogue is not a speech tag, and therefore a period should replace the comma after think.

[“Actually, there might not be,” the man agreed after thinking for a few moments, “Sometimes my drinking friends would call New York a city of dreams.]
The comma after moments should be a period - if they're both, however, the same sentence, (but I don't think they are/should be), then sometimes would be uncapitalized.

[drive in an out]
"an" should be "and."

[They would laugh amongst themselves, but would never care to tell anyone in the garage the joke, “What did you find there?”]
The part before the dialogue is not a speech tag, so the comma after joke should be a period.

[The handsome city man only smiled casually, as though there was some great secret about the city that he didn’t want to spoil, “Where’s your Pops, Kid?”]
The part before the dialogue is not a speech tag, and therefore the comma after spoil should be replaced with a period.

[“Where’s your Pops, Kid?” he asked, changing the subject, “You said he ran this joint, right?”]
The comma after subject should be a period.

[I shook my head, “Not sick. In the city,” I corrected, “She took a train to the city and has been there for a day and a half.”]
Shaking your head does not produce words, and hence it is not a tagline; that should be a period after head. In this case, the following dialogue (starting with she) is continuing from the first, so "she" shouldn't be capitalized.

[The man nodded, his eyes turning pretty misty, “I better get going,”]
Nodding and eyes turning misty is not a tagline (by the way, even if they were, you follow up with another tagline, so it would be incorrect anyway), and therefore that should be a period after misty.

[I began to feel a little sick in my stomach, suddenly knowing how my father must have felt, “Been happening a lot lately.”]
That should be a period after felt, and not a comma, because it is a tagline.

To sum up any further grammatical errors, I suggest checking up on dialogue punctuation (it's mostly just one repeated mistake).

[I shook my head solemnly, “No Sir.]
That comma should be a period. Comma before Sir, also.