Welcome to the forum. Logic can be considered as a part of discrete mathematics (since traditionally there are just two truth values 0 and 1 and not the whole continuum between 0 and 1), so logic-related questions can be posted to the Discrete Math subforum.

(a) Suppose T is a set of axioms and A is another axioms derivable from T. Then everything derivable from T is also derivable from T with A, using the same derivation. Conversely, if a derivation from T plus A uses A at some point, then we can construct a derivation from T only, which first derives A and then uses it as necessary.

Suppose M is a model of T plus A; then M is a model of T only. Conversely, suppose M is a model of T. Since A is derivable from T, the soundness theorem says that every model of T is also a model of A, so M is a model of T together with A.

(b) Suppose that T consists of independent axioms and for some A ∈ T, let A' be A /\ (p -> p) (a conjunction of A and some tautology). Then A and A' are derivable from each other, i.e., they are dependent. If we remove both A and A' from T plus A', then we are left with T without A, which is weaker than T because axioms in T were independent (in particular, A was not derivable from other axioms).