Since $g(f(x))=x$ for all $x\in\mathbb{R}$, the function $g$ is the inverse of $f$.

Hence, $g=f^{-1}.$

We need to find $\dfrac{g(7)}{g'(7)}$.

First, find $g(7)$. Since $g$ is the inverse of $f$, this means we need the value of $x$ such that $f(x)=7.$

Compute $f(1)$: $f(1)=1^5+2(1)^3+3(1)+1=1+2+3+1=7.$

Therefore, $g(7)=1.$

Use the derivative formula for an inverse function: $g'(y)=\frac{1}{f'(x)}\quad \text{where } y=f(x).$

Here, $y=7$ and the corresponding $x=1$, so $g'(7)=\frac{1}{f'(1)}.$

Differentiate $f(x)$: $f'(x)=5x^4+6x^2+3.$

Now evaluate at $x=1$: $f'(1)=5+6+3=14.$

Thus, $g'(7)=\frac{1}{14}.$

Now compute $\frac{g(7)}{g'(7)}=\frac{1}{1/14}=14.$

Therefore, the correct option is $\boxed{14}.$