Would a faraday cage still work if it was part of a circuit?
Note: Every conductor has resistance. The current in a lightning strike is very large (thousands of amperes). If the Faraday cage is not sufficiently conductive, there may be a voltage drop between where current hits the cage and where it leaves. That voltage drop could cause some current to flow through alternate paths inside the Faraday cage. For the purposes of this answer, I will assume that the Faraday cage is sufficiently conductive to protect it's interior from currents that are likely to strike it. If it is not sufficiently conductive, well, all bets are off.
If a circuit consists only of components completely inside of the Faraday cage, plus the Faraday cage itself, then the components inside of the Faraday cage will be protected from electrical discharges originating outside of, and some distance away from, the cage.
The cage could be used, for example as the &quot;ground&quot; of a circuit, as is often the case in airplanes
However, if the circuit has any part that extends beyond the cage, then this protection will be lost. For example, a circuit in a Faraday cage which has a ground wire that extends outside of the cage to earth, (but is not bonded to the cage itself) would not be protected. Potentially, a discharge could pass from a source (such as lightning) to the cage, then jump to the ground wire (possibly through circuitry inside the cage) and continue to the earth.
On the other hand, a circuit which is entirely enclosed, but contains a battery or other electrical source of power, will have protection. For example, the electronics in airplanes struck by lightning are typically unaffected. (The &quot;Faraday Cage&quot; of a commercial airplane is designed to be highly conductive, and tested for the currents it is likely to experience in a lightning strike.)

Edit: The same principle applies to changing electromagnetic fields. If a circuit contained within a metal box is adequately isolated from external electromagnetic fields by that box, then including that box into the circuit, for example by making it a ground &quot;net&quot;, will not compromise the isolation afforded to that circuit.
Nor will driving a current through a Faraday Cage from an external source adversely affect anything inside the Faraday cage (of course always with the proviso that the cage is adequately conductive so that no voltage develops across the cage, and there is no appreciable heating of the cage as the current flows through it.)