Is the halting problem decidable by an "infinite Turing machine"?
It is true that infinite Turing Machines as you have defined them can solve the halting problem for regular Turing Machines. That is, there exists an infinite TM, $I(t,x)$, that returns $1$ iff $t(x)$ halts. To see this, just have your first step be “run t(x) for infinitely many steps” and the rest is easy after that. Yes, there are some issues formalizing this idea, but it can be done. This is called a “Hyper Turing machine”

There’s an interesting question lurking here though... is there an infinite TM that solves the halting problem for infinite Turing Machines? The answer to that is no, and if you know the diagonalization proof of the incomputability of the halting problem that applies here too.