Regarding computability yes. Regarding time complexity, it is unproven.

What we do know is that there is a known algorithm to factor number in polynomial time on quantum computers and there is not a known algorithm to do so on classical computers. Assuming that factoring numbers is actually a hard problem on classical computers (unproven, but generally accepted), then quantum computers are more powerful than classical ones.

I'm not sure what you're referring to; everything in the parent post is correct. You are right that quantum and classical computers have the same "power" in terms of problems that can be solved, but it is also true that quantum computers may be able to computer certain functions faster than classical computers (using faster algorithms that don't run on classical computers).

I'm referring to "They allow the running of some algorithms that normal computers can't physically run."

I understood this as, it's impossible to run some algorithms on a traditional computer. You can run Shor's algorithm on a Turing machine, albeit requiring a simulation of qbits. You can run it in javascript and in your browser if you like.

True, you can simulate the qbits, so the more precise statement would be they can execute some algorithms in a way that classical computers cannot and this gives them superior efficiency in some situations.

You're right. I should have specified that they can run certain algorithms with a runtime complexity impossible on a classical computer.

If you take into account the probability of breaking a given 4,096-bit RSA cipher before the heat death of the universe, it is correct. However this is an implementation detail, and a hardware problem to boot :p

Rather mathematical response to a question requesting an answer for non mathematicians ;-)

Since Moore's Law is giving us one more bit of decryption per unit time every 1.5 years, 4096 * 1.5 years is probably going to precede the heat death of the universe handily. But 6000-or-so years of Moore's Law does tax credulity.