When valves are soldered to copper pipe, the thicker walls require more heat to transpire. Copper sweating is similar, where it rubs against the outside of the pipe and the “cup” of the fitting to be soldered. Melt both and heat the area around the mug. Then place a torch flame in one spot, applying solder to the opposite side until the heat causes the solder to flow into it. Solder will flow hot without exception. But overheating is a possibility, so remove the torch from the pipe once the solder flows into the heat and wipe it off immediately with a damp rag.
Cut cast iron and plastic/cast connections? Cut the iron box with a reciprocating saw, maybe a Milwaukee saw, the whole thing. Cast iron is very strong, so you’ll want to use heavy metal blades. I like the Lenox 614R and start with six-inch blades. It will take some time to penetrate the side of the pipe, but it will go faster once there is a hole there. Longer metal blades are needed to finish the job. It may seem like it won’t work, but trust me, it will. To connect plastic and cast iron, use Mission or Fernco No Flex couplings. Make sure you don’t forget the vents.
Add: When looking for iron drain pipes, I like to use a rigid ratcheting cast iron cutter. However, sometimes the area is too limited to use that tool. In this case I use a mini grinder and a diamond blade to cut as much as I can. The diamond wheel costs $100, but one will last you forever. I would end up with a hacksaw and grating edge blade. By the way, the diamond disc grinder is ideal for cutting tile, concrete and brick. It will be quite dusty to cut these things.
Adding a new drain to a cast iron pipe where there is a cleanout? There are a couple of ways to do this. The brass “little rect.box” is a cleaning plug. Unscrew it if you can, but if not, cut out the square with a hacksaw, the whole thing. Then cut from the center towards the strands into pie-like sections. Remove the plug from the female threads. Ideally it will be three inches. Screw on a three-inch male adapter. But it could be 3 ½ inches, which is no longer in production. Use a Fernco 5×3″ hood with the 5″ over the cleanout bucket. Push down with a 3″ by 2″ socket on the 3″ end of the Fernco. Another way to do this is to take out a section of the cast pipe and use Fernco around a PVC or ABS sanitary tee. 6″ is most likely a 4″ cast. It can be cut with a saw or broken with a ratcheting cutter designed to penetrate cast iron. You may want to rent or borrow one as they usually cost more than $300. When taking sections out of the pipe, it is important for safety reasons that the upper section of the pipe be supported.