So how does Gil travel through time? Is it the clock, the street?
The car. Each time he goes back to the 1920s, he does so by accepting ride in an antique car. When he and Adrianna go to La Belle Epoque, they do so by accepting a ride in a horse-drawn carriage from the period.
How the Hell did Gil manage to find the diary of a nobody from ninety years ago at a street stall in the middle of Paris?
Not much of a nobody.
How could Gil pay for his drinks in 1920? The money is different and all.
Perhaps everybody was paying for him, since he was like "the tourist" and "their guest" or whatever.