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I would say that it should take around 30 days for the FHA streamline refinance. Since you have all your information together from the last lender you should be able to easily give that information to another lender if you wish to do so. [self promotion deleted by Zillow moderator.
There are big profits in marking up customers finance rates! As for how long before you can refinance your car loan, you'd want to be sure to review the contract you signed to see if there are any prepayment penalties. More than likely there are not and you should be able to refinance at any time. Many dealers will tell you to wait 6 months and this is sometimes true if someone is trying to build credit, but is not the case if your credit is already OK.