lo.logic - Propositional Logic, First-Order Logic, and Higher-Order Logics

I've been reading up a bit on the fundamentals of formal logic, and have accumulated a few questions along the way. I am pretty much a complete beginner to the field, so I would very much appreciate if anyone could clarify some of these points.

1. A complete (and consitent) propositional logic can be defined in a number of ways, as I understand, which are all equivalent. I have heard it can be defined with one axiom and multiple rules of inferences or multiple axioms and a single rule of inference (e.g. Modus Ponens) - or somewhere inbetween. Are there any advantage/disvantages to either? Which is more conventional?




2. Propositional (zeroth-order) logic is simply capable of making and verifying logical statements. First-order (and higher order) logics can represent proofs (or increasing hierarchial complexity) - true/false, and why?




3. What exactly is the relationship between an nth-order logic and an (n+1)th-order logic, in general. An explanation mathematical notation would be desirable here, as long as it's not too advanced.




4. Any formal logic above (or perhaps including?) first-order is sufficiently powerful to be rendered inconsistent or incomplete by Godel's Incompleteness Theorem - true/false? What are the advantages/disadvantages of using lower/higher-order formal logics? Is there a lower bound on the order of logic required to prove all known mathematics today, or would you in theory have to use an arbitrarily high-order logic?




5. What is the role type theory plays in formal logic? Is it simply a way of describing nth-order logic in a consolidated theory (but orthogonal to formal logic itself), or is it some generalisation of formal logic that explains everything by itself?

Hopefully I've phrased these questions in some vaguely meaningful/understandable way, but apologies if not! If anyone could provide me with some details on the various points without assuming too much prior knowledge of the fields, that would be great. (I am an undergraduate Physics student, with a background largely in mathematical methods and the fundamentals of mathematical analysis, if that helps.)