Also, is this something that has to be resolved before big advancements can be made in strong AI?
It is not provable and it doesn't matter. It really doesn't matter if an AI is really thinking or not - all that matters is can it simulate thinking to a level that we can't tell the difference.
Given that you can emulate a turing machine in your mind, I'd assume that it is more than a Turing machine.
But no infinite tape. I can't remember if that's essential to the definition of a Turing machine?
Obviously no physical machine can have an actual unbounded tape (note that unbounded is not the same as infinite!) The interesting question is whether the brain is Turing-equivalent if given a notebook with an unlimited number of pages to keep track of the program state.
Any Universal Turing Machine can emulate any other Turing machine. That's not interesting. What would be interesting would be if human minds could solve the Halting Problem.
This would prove that humans are not Turing machines - so far no human has solved this.
I don't think this is actually known.
So then it has the power of (at least) a universal turing machine.
It is not proven that a human mind can emulate all Turing machines (given an arbitrary amount of external storage). All physical computers are "just" finite state machines (because they don't have unbounded storage); you can emulate a subset of Turing machines using an FSM just fine even though a TM is strictly more powerful than an FSM.