The recurring line "But if you want to stay around and love me" is sung to far fewer pitches than the surrounding lines (it simply alternates between Eb and F), and in a way, this narrow scope reflects the "stay[ing] around."
That the first verse and the third verse use the same sound for their rhymes ("door" with "floor" and "poor" with "door," respectively) poetically represents the narrator's limited means.
The phrase "mess around" in the first line may indicate Ray Charles' influence, specifically his "Mess Around," in which this phrase frequently recurs. "It's Alright with Me" is in C minor, and if I'm not mistaken, "Mess Around" is in Eb major, so the two songs are in relative keys.
