Simulating Turing machines (output included) with circuits
Yes, as long as we allow encoding the the result of $f$ in a basically trivial way, rather than outputting exactly $f(x)$.

To get over obstacle (2) (different output lengths for the same input length): for any given input length $n$, there is a maximum output length $m$.  $C_n$ will output $2m$ bits.  The first $2|f(x)|$ bits output by $C_n(x)$ will be the bits of $f(x)$ doubled, and the remaining $2(m-|f(x)|)$ bits will be the string 01 repeated over and over.  For example, if $m = 6$ and for some $x$ of length 10, $f(x)=0010$, then $C_{10}(x) = 00\ 00\ 11\ 00\ 01\ 01$ (spaces added for ease of reading).

To get over obstacle (1) ($f(x)$ may not halt): for any given input length $n$, there is a time $t$ such that, for all $x$ of length $n$, either $f(x)$ halts within $t$ steps, or $f(x)$ does not halt ever.  If $f(x)$ does not halt, then we make the output of the circuit be a string of 10 repeated over and over again.  Of course, such a circuit family can not in general be uniform, since deciding that halting time is in general uncomputable, but the circuit family still exists.