Are there any restrictions against giving gifts to new credit card customers? This would go only to new account holders. Applicants that applied, but did not receive the new account would receive nothing - other than an adverse action letter. Would it be the same as giving a new customer a discounted origination fee?

Other than RESPA restrictions there are no loan payment restrictions. Even RESPA, in this instance, would be moot as this is between the lender and customer. There is no "bonus" disclosure in Reg. Z comparable to Reg. DD on the deposit side.