How to replace a section of cast iron drain pipe

For cutting the pipe, go rent a chain pipe cutter (aka soil pipe cutter). It will make short work of the pipe and not be too messy. It will make a clean enough edge that a Fernco coupling (like you have a picture of) will work fine.

Obviously you will need to add some strapping to secure the horizontal run of iron pipe if you go this route because you don't want to put any extra strain on that rubber coupling.

I would not mess with trying to remove the flange. I think it will be easier to just cut the pipe.

No, there should not ever be two traps on a single drain line.

The point of the trap is to create a water barrier which prevents sewer gasses from coming up the empty pipe and into the house.

When you have two traps on the same line, you end up creating a kind of vapor block in the line which prevents it from draining properly.

What do you do?

Get yourself some 1 1/2" PVC (not CPVC, which is yellow, PVC which is white) with a 90 deg elbow, and also get two Fernco couplings (black plastic couplings with straps on each end that can be tightened) that will fit the pvc and iron drain pipe.

Use a metal blade on a sawz-all and cut out the p-trap in your basement. (Have a bucket handy!) Leave room to connect the Fernco coupling on both sides. I'd cut the right hand side out just past the coupling.

Measure and cut the PVC so that with the 90 elbow the ends line up with the newly cut drain lines. Glue the PVC together and confirm the fit, then connect to the old drains with the Fernco.

Viola - a proper drain line that will drain smoothly w/ no vapor lock.