Converse non-implication - converse non-implication is like basically saying that the consequent does not imply the antecedent. ~(P <-/- Q) This is not precise logical notation though. It's like a warped half DeMorgan.

Material non-implication - it is basically like saying the antecedent does not imply the consequent. ~(P -/-> Q). It's like a complicated negation of a conditional with a flipped truth value.

As for what you can infer from them? These are very warped versions of replacement rules. There is an inference pattern, but I have never used them before because of better replacement rules.