Clarification of material conditional, logical necessity and causation

Question

Are the following four statements always true -

1. If Proposition B is the Logical consequence of proposition A, then B is material conditionally connected with A.
2. If Proposition E is material conditionally connected with C, then it is NOT necessarily the case that E is a logical consequence of C.
3. If proposition H corresponds to an observed fact H* which is supposed to be causally related to observed fact G* (say H* causes G*), then the proposition G corresponding to the observed fact G* is material conditionally connected with Proposition H.
4. If proposition X is material conditionally connected with Proposition Y, then it is NOT necessarily the case that X and Y are causally related.

I am a student of Physics, researching on quantum logic. I am confused by the above statements which I hypothesized regarding classical logic. Any help would be much appreciated.

Definitions taken from the comments:

Material conditional: Proposition A is material conditionally connected with Proposition B, denoted as "If A then B" when the proposition "If A then B" has a truth value of false only if A is true and B false. For all other combinations of truth values assigned to A and B, "If A then B" is true.

Logical Necessity: Proposition A is a logical consequence of proposition B, if the truth of proposition B (along with maybe other auxiliary axioms), necessitates the truth of Proposition A. Example: The fact that an equilateral triangle has all three sides equal necessitates that all of the three angles are each 60 degrees.

Answer 1

On material implication/conditional: A => B means only that it is not the case that both A is true and B is false. Or equivalently, A is false and/or B is true. Nothing more. No connection (causal or otherwise) is assumed between the propositions A and B.

Note: While the above is often stated as a definition of material implication, it can also be derived as a theorem from simpler, self-evident properties of logical connectives including '=>' itself.

See my recent blog posting, "Material Implication: If Pigs Could Fly". There, I attempt to formally justify each entry of truth table and other well known properties of material implication.