Evapo-rust is an excellent rust remover for this purpose. I have used it for successful restoration of old planes. I prefer mechanical removal, but that requires specialized equipment that is not worth the investment for a few planes. DO NOT USE A BELT SANDER! Granted a large plane like a No. 7 IS going to be a lot of work to hand lap, but there are many reasons to not use a sander. As you mentioned, your sander is not flat enough. The contact bed is also only about 4” vs the 21 1/2”-22” sole of you plane. Most importantly, the belt sander will rapidly, and unevenly heat the sole of the plane causing measurable distortion. You are then removing material out of “plane” with a flat sole. As the plane cools, it returns to original position and you now have low spots that are even more difficult to remove. As soon as the plane is warm to the touch from a power sander, you are working on distorted sole.
Lap the sole with abrasive paper on a flat surface, table saw, glass plate, polished granite (check that these are flat to your desired outcome). Best to have knob, tote, and frog installed. I would disagree with the idea these items must be installed for the plane to be flat under tension, but they make good handholds for lapping. See how out of true it is, then determine how to correct it. Once you determine you have areas that require attention, work with course abrasive, 80 grit or courser (I have used 30 grit when necessary), until you have full contact. Then proceed to finer grits, small jumps, until you reach your desired finish. For most this is 120 grit (you can polish to what ever level you prefer. I am a nut, and polish plane soles and cheeks to 8000 grit or higher, producing near mirror finishes that I enjoy and that are less rust prone). Each time you move to finer grit, you must have full contact with preceding grit or you will spend an enormous amount of time trying to remove the marks from the prior step.
Porter-Cable makes some fantastic, self adhesive sandpaper for the task. You will have more than you need for several planes, but the roll paper is much easier for working on larger planes. Available in various grits.
For finish, the closest you can get to original japanning is Duplicolor Ford Semi-gloss black, DE1635. Available at most automotive supply stores. Use a good quality, black primer first. If you want a smooth finish, you will have to remove any remaining, original japanning. Household paint strippers may do the job.
Remember, you are working on a tool that has already lasted two lifetimes, and will serve several more if you invest the time and effort. Stripping the plane of rust and finish will take several days. Lapping may take a total of 3 hours over the course of a few days (depends how you feel about mindless repetitive work (or meditative, depends on the person). Flattening and honing the cutting iron, a couple hours, lapping and fitting the chip breaker, less than an hour. The joy of running a beautifully restored, well tuned No. 7 plane, lasts a lifetime. And the next person’s lifetime as well.