In your picture, you have 3 rubber couplings near the top. The two in the foreground, which have no overall metal shield, are for underground use only, and commonly called "ferncos". The one in the background does have an overall metal shield, and if it is made by the company Fernco, their product line is called "Proflex."
So for connecting to the cast iron, your options are a donut or a Proflex. And once you figure that out (no experience on my end, so no opinion), do replace all of the copper and replace the "ferncos" with Proflexes or with glue joint connections. Seems like if you demo everything from the Proflex/ferncos down to the cast iron, you could reconstruct with more ABS and just one or two Proflexes total.
Also, your washing machine needs a standpipe, a trap, and a vent, which could be an AAV in your jurisdiction. So fix that when you rebuild the DWV.
Cheers, Wayne