Proofs using Rules of Inference

Question

I am asked the following question:

As you can see, I am tasked with trying to find the discrepancies in the proposition and the steps of proof. The two discrepancies I've found were the following:

Step 4: Using simplification on the premise H gave us P(x).

Step 6: Using existential generalization on P(c) ∧ Q(x) would be ∃xP(c) ∧ ∃xQ(x)

But I'm not sure if I am correct.

Answer 1

From Step$2$ and Step$4$ what it get is actually:

Step$2:P(c_1)$

Step$4:Q(c_2)$

So Step $5$ become $P(c_1)\land Q(c_2)$

Since there is no guarantee of $c_1=c_2$, which means apply existential generalization won't imply step $6$

Therefore that proof is not valid.