Should I remove the old flare from the pipe or from the nut at the shut-off valve? I can't (or shouldn't) connect the new line to the old flare, right?45839

That gas valve is not aga approved. Per code you must replace it. What kind of fitting you are using for your gas connector will determine weather you need male or female threads.
Mac

Code would state a lot of things here...

It would tell you to replace the gas shut off valve.

It would tell you install a new 3' flexible connector.

It would tell you that you need to pull a gas permit and test your work for a plumbing or gas inspector.

Code would also tell you that you as a homeowner cannot TOUCH the gas piping...;)

Now, with all that being said, if you are NOT going to hire a plumber or gasfitter to do the work and you are determined to do this job on your own then you would leave the gas cock (tighten nut on back side a 1/2 turn when you turn gas back on) and remove that old flare fitting and install the new flare fitting that comes with the new flexible gas connector. Connect everything, turn gas on, tighten nut on backside of the old gas valve and then use a soap/bubble solution across all threads and the shutoff valve to test that your work is gas tight and has no leaks.

Finally, most important with any stove installation will be for you to install the ANTI.-TIP DEVICE that comes with all stoves. This device saves lives, so be sure to install it, OK?

Back to you...

Mark