1. We are given $g(x)=h(e^x)\,e^{h(x)}$ with $h(0)=0,\\ h(1)=1,\\ h'(0)=2,\\ h'(1)=2.$

2. We need to find $g'(0)$.

3. Differentiate $g(x)$ using the product rule: $g'(x)=\frac{d}{dx}\big(h(e^x)\big)\cdot e^{h(x)}+h(e^x)\cdot \frac{d}{dx}\big(e^{h(x)}\big).$

4. Now apply the chain rule to each derivative:

   • For $h(e^x)$, $\frac{d}{dx}h(e^x)=h'(e^x)\cdot e^x.$
   • For $e^{h(x)}$, $\frac{d}{dx}e^{h(x)}=e^{h(x)}h'(x).$

   So, $g'(x)=h'(e^x)e^x\,e^{h(x)}+h(e^x)e^{h(x)}h'(x).$

5. Factor out $e^{h(x)}$: $g'(x)=e^{h(x)}\left(h'(e^x)e^x+h(e^x)h'(x)\right).$

6. Substitute $x=0$: $g'(0)=e^{h(0)}\left(h'(e^0)e^0+h(e^0)h'(0)\right).$

7. Use the given values:

   • $h(0)=0 \implies e^{h(0)}=e^0=1$
   • $e^0=1$
   • $h'(1)=2$
   • $h(1)=1$
   • $h'(0)=2$

   Therefore, $g'(0)=1\left(h'(1)\cdot 1+h(1)h'(0)\right)=2+1\cdot 2=4.$

8. Hence, $\boxed{g'(0)=4}.$

9. Comparing with the options:

   • A: $4$ ✅
   • B: $5$
   • C: $3$
   • D: $8$

So the correct option is A.