Relations between statements involving universal quantifier, conditional and biconditional
Reasoning based on examples is fine for disproving implications, as you have done. In order to prove that an implication holds, however, you need something more elaborate. One possibility, as mentioned in the comments, is to use resolution to prove the implications. For example, suppose you want to show that (a) implies (b), i.e. that

$\varphi:=(\forall x.(b(c)\to c(x)))\to((\forall x.b(x))\to(\forall x.c(x)))$

is valid. We will do this by deriving a contradiction from $\neg\varphi$, which (using standard equivalences) is equivalent to

$\forall x.(\neg b(x)\vee c(x)))\wedge\forall x.b(x)\wedge\exists x.\neg c(x)$.

Skolemization results in 

$(\neg b(x)\vee c(x))\wedge b(x)\wedge\neg c(a)$,

from which resolution derives first $\neg b(a)$ and then an empty clause, i.e. false. Therefore the original implication is valid.

As for your table, note that the example I gave also shows that (d) does not imply (a); the remaining 5 implications look ok.