Would there be a significant difference if a metal shield is connected or not to the circuit ground?
First you should know that metal shielding is for protection against stray magnetic fields(Electromagnetic Interference, which induce eddy currents in wires/conductor). In the first part of your question, there is no need to connect your metal shield to the ground. You can if you want to, but ground loops are generally avoided if you have long wires. In the second scenario, you just have to buy wires that are either co-axial or twisted pair. Both are optimized for minimizing external interference in the signal they are carrying as well as minimizing the magnetic fields they are providing.
Also, avoid switching ac to dc power supplies, they are also one of the main causes of electromagnetic interference. But if you do have to work with them. Use a three terminal prong to have a ground/earth reference (it alleviates the problem to some extent) and use a voltage regulator at its output so that ripples are reduced. (Adding external Capacitors and inductors requires alot of experience and fine tuning so voltage regulator is your best bet).
you can also use other components such as an opto-isolator at the ADC output so load and external factors do not affect its output.