Is this formula bellow
$$\lnot(\forall a,b,c \in X: (aRb \wedge aRc) \Rightarrow \exists d (bRd \wedge cRd))$$
equivalent to
$$\exists a,b,c \in X: (aRb \wedge aRc) \Rightarrow \not\exists d (bRd \wedge cRd)$$
and does the second part of the formula mean that an exclusive or(or nothing at all) is possible here namely that we have $cRd$ or if we don't have $cRd$ we can have $bRd$ or if don't have that b and c relate to no other is possible?
They are not equivalent.
The negation of: $$\forall a,b,c\in X\;[p\to q]$$ is: $$\exists a,b,c\in X\;[p\wedge\neg q]$$
Note that $p\wedge\neg q$ and $p\to\neg q$ are not equivalent.