I hope this doesn't come across as picking holes in the question.
The first point, as raised by dkuper, is that ordinary circuits necessarily implement total functions (every input gives either true or false), whereas Turing machines implement partial functions (inputs lead to acceptance, rejection or "no answer", i.e., non-termination). If you want to simulate a Turing machine, you're either going to need to add something to your model of circuits that allows them to compute partial functions, or implement separate circuits for, say, "The machine accepts this input", "The machine rejects this input" and "The machine doesn't halt on this input."
The second point is, what does it mean to have a circuit with infinitely many gates? Any circuit with an infinite number of gates and a single output gate must have an infinite path and/or a gate with infinite in-degree. What does it mean to evaluate an infinite path? Can that be done in finite time? Is it OK to have a gate with infinite in-degree, given that such a thing couldn't possibly be realised? (OK, the infinite tape of a Turing machine can't be physically realised but it suffices to have a finite amount of tape and promise to add more whenever the head reaches the end.)
Now let's look at the more specific questions. The version where the Turing machine $M$ and its input $x$ are fixed, is trivial, as are all problems with a finite number of instances (one, in this case). The circuit is the constant "true" if $M$ accepts $x$, the constant "false" if it rejects and, some circuit with no value if it doesn't halt (or whatever you decided to do to represent non-terminating computations in your circuit model). We know that this circuit exists but, of course, the function that maps $M$ and $x$ to the required circuit is uncomputable.
Finally, the case where the circuit's input is (a coding of) the Turing machine and its input. You have to decide how to code that as the input to an infinite circuit. One way would be to code machine and input as a finite binary string and have the infinite sequence of inputs $x_i$ and $y_i$ ($i>0$) to the circuit such that, the circuit evaluates to false if every $y_i=0$ and, otherwise, putting $\ell=\min\{i\mid y_i=1\}$, the sequence $x_1\dots x_{\ell}$ is the binary string coding the input. Now, take the infinite set $S$ of pairs $(M_j,x_j)$ ($j>0$) such that $M_j$ is a machine that accepts input $x_j$. The required circuit is just an infinite disjunction, where the $j$th disjunct says "My input codes $(M_j,x_j)$"; again, you need to deal with non-terminating computations in some way.
Alternatively, you go the more conventional route of having a family of circuits, one for each input length, as described by dkuper.