Why doesn't Fagin just report to the police that there's a madman threatening him for money?
Well, because he would probably have to confess that he made deals with a loan shark, which is not exactly legal. That and the fact that Fagin is a thief (albeit a harmless one) must have been what kept him from seeking the police. It´s also possible that Sykes could have killed the policeman sent after him and for Fagin, it would only make things worse if Sykes found out he was the one who tried to get him arrested.
Sykes is heavily implied to be high up in the Mafia. That's not the kind of person you snitch on without repercussions.
How'd Dodger get his bandanna back after the climax?
Fagin probably got him a new one. What I wanna know, is where the heck did his bandanna go during the "Why Should I Worry?" sequence?
The "Why should I worry sequence" was just a simple Animation error (Movie goofs; they happen. They're all over IMDb)
Why would Sykes even bother loaning cash to a bum like Fagin in the first place? Assuming he was always homeless, did Sykes really expect Fagin to pay him back?
Sykes doesn't care that Fagin's poor. He knows the old bum has got to pay him somehow.
The trouble with borrowing money from Loan Sharks is that you tend to end up even poorer than when you started, with fun things like 500% interest per week. Fagin may not even have been homeless until he started borrowing money from Sykes.