The question was essentially answered by alpoge.

For the first question note that the function from the absolute Galois group of $\mathbb{Q}$ to $\mathbb{Q}_l$ sending an automorphism to the trace of the matrix it acts by on $H^i_c$ is the same for the Frobeniuses of all but finitely many primes. The Frobeniuses of all but finitely many primes are in fact a dense subset of the absolute Galois group so two continuous functions agreeing on them must be equal. Then the traces of the powers of Frobeniuses are also equal.

For the second question the Brauer-Nesbitt theorem (formulated for profinite groups) shows that $H^i_c$ have isomorphic semisimplifications and then we would be done if we knew that smooth proper varieties have semisimple cohomology. But the latter is a hard conjecture so this is where we are.

The question was essentially answered by alpoge.

For the first question note that the function from the absolute Galois group of $\mathbb{Q}$ to $\mathbb{Q}_l$ sending an automorphism to the trace of the matrix it acts by on $\sum\limits_{i\geq 0}(-1)^i[H^i_c]$ is the same for the Frobeniuses of all but finitely many primes. The Frobeniuses of all but finitely many primes are in fact a dense subset of the absolute Galois group so two continuous functions agreeing on them must be equal. Then the traces of the powers of Frobeniuses are also equal.

For the second question the Brauer-Nesbitt theorem (formulated for profinite groups) shows that $H^i_c$ have isomorphic semisimplifications and then we would be done if we knew that smooth proper varieties have semisimple cohomology. But the latter is a hard conjecture so this is where we are.