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If the rust is not cut out and new steel welded in and painted, the rust will continue to spread. But I assume you're not trying to make this car last 10 more years, but just get you by, keep things dry and go a few more years.
Your idea will work fine. You can get sheet metal at Home Depot or Lowe's in small pieces. You should buy some silicone in a tube that will fit a caulking gun (gun itself is very inexpensive.) It would be nice to bend one edge of the sheet metal up so it will screw to some fairly solid metal under the driver's seat. A vise would help here, but some good pliers may be adequate if you don't get really heavy-gauge sheet metal.
If you have a cordless screwdriver, buy phillips-head sheet-metal screws and you probably won't even have to drill any holes, as the screws will go right through the new piece and the car body. Be sure to caulk with the silicone completely around the edge to seal out water before you screw things down, and again caulk the edge of the metal after you're done. Use plenty of screws, as they'll begin to lose their effectiveness as the floor continues to rust away.
If you had access to a small angle-grinder you could put a cut-off wheel on it and cut the excess rusted metal away. Those pieces peeling away underneath will continue to rust and spread. There are some rust-inhibiting paints that claim to stop rust, and you might use one of those to try and slow the rusting process before you make your repairs. Good luck!
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