swear by the sin of Samaria - namely, the calves (Deu 9:21; Hos 4:15). “Swear by” means to worship (Psa 63:11).
The manner - that is, as “the way” is used (Psa 139:24; Act 9:2), the mode of worship.
Thy god, O Dan - the other golden calf at Dan (1Ki 22:26-30).
liveth ... liveth - rather, “May thy god ... live ... may the manner ... live.” Or, “As (surely as) thy god, O Dan, liveth.” This is their formula when they swear; not “May Jehovah live!” or, “As Jehovah liveth!”