This may depends on the context, as there is not a single logic based on lambda calculus. Ex. two of the most famous are Higher Order Logic and Calculus of Constructions.
An axiom is a starting point taken for granted. An inference rule is a rule by which, from an (or multiple) axiom/theorem/lemma, you can produce a new theorem or lemma, which is a proven axiom (otherwise an axiom may be supposed valid, while not proven to be so… an axiom is a starting point).
From that follows the axioms of lambda‑calculus are the syntactic rules (there also are “syntactic proofs”) and interpretations of formulas (note interpretations may vary, depending on the context) and inference rules are the rewriting and transformation rules, ex. application, abstraction, substitution, beta‑reduction and some others. Also note inference rules, are just special kind of axioms.
According to Introduction to Lambda Calculus [pdf] (Henk Barendregt, Erik Barendsen), the axiom of substitution, is the only required axiom (an inference rule is often taken for an axiom in the logician world).
For the semantics, it depends on the interpretation: denotational semantic and operational semantic, are both valid options, while two different enough options.
I'm posting this so that you at least get an answer, but please, note you can expect a better answer than mine, as I'm not a lambda‑calculus guru.
Worth reading too:
A nominal axiomatisation of the lambda-calculus [pdf] (Murdoch J. Gabbay, Aad Mathijssen)
The lambda-calculus is fundamental in computer science. It resists an
algebraic treatment because of capture-avoidance side-conditions.
Nominal algebra is a logic of equality designed for specifications involving
binding. We axiomatise the lambda-calculus using nominal algebra,
demonstrate how proofs with these axioms reflect the informal arguments on
syntax, and we prove the axioms sound and complete. We consider both
non-extensional and extensional versions (alpha-beta and alpha-beta-eta
Worth to be noted in this context, to avoid confusion: the lambda calculus on it's own, is not a valid deductive system, and this is demonstrated by the existence of a function named the fixed‑point combinator , which introduce a paradox of one of the most common type: self‑referential paradox. However, some part of it, or at least of some kind of lambda‑calculus, may be sound. Ex. the type system of at least SML (I don't know for other concrete system of lambda‑calculus) is sound. In this example context, providing types are assigned an interpretation (which is straight‑forward with the Curry‑Howard isomorphism where a type is a logical proposition), the type system (if I understand correctly), is a sound deductive system. As an example, HOL systems (the proof checkers and proof assistants family, not the logic of the same name they are based on), rely on SML's type system to formally ensure the validity of the inferences sequence they check.
 As a reminder, the fixed‑point combinator is the function of this canonical form (there are other way to express it though):
val rec fix = fn f => (fn x => (f (fix f)) x)
Or more succinctly:
fun fix f x = f (fix f) x