So I believe this is a very important question. Sir, you are on right path, you should run transient and AC simulations. So basically, you should have a periodic operating point instead of a DC operating point right? Virtuoso has a great tool for that, which are the pss and pac functions.
Pss simply runs a transient simulation with a periodically changing operating point. So what do you expect from the results? It is simple, since you only calculate the harmonics of fundamental freq and switching freq when using pss, you only get the frequency components at those frequencies. It is important to note that this type of analysis will reduce the simulation time significantly, provided that the switching circuit settles quickly. Otherwise it is going to integrate on and on without starting the real pss analysis.
Anyway, I believe this wasn't what you are looking for. What you are looking for is doing AC analysis for a periodically changing operating point. PAC does that. When you run PAC you will get your gain, also there is a function called pnoise which can help you with noise sims.
So, this is the easiest way to do if you have a powerful tool like cadence virtuoso. There are many other ways to do the measurements you mentioned, but to my knowledge this is the most reliable and fastest.
Edit: Fully dif opamp without CM feedback might not work. Even if it is ideal it might not give you meaningful results.