Is it safe to keep driving my Wrangler with the floor rusted through?

The seat mount is the part that should scare you, not the hole itself. If the seat bolts are anchoring into rusty sheet metal, that's a hard no on driving it, even around the block. In a crash the seat only has to move an inch or two to turn a survivable hit into a bad one, and your left foot bracing on soft floor while clutching is its own problem. Tow it or park it until the floor's done. As for the repair itself, yes, this is totally doable in a driveway. I've patched two TJ floors on my carport with a cheap flux core machine. The trick is being aggressive with the cutoff wheel. Cut back until you're into bright, thick metal with zero flakes or scale, and don't trust anything that looks solid but has that orange blistering underneath. On mine the rot ran further toward the rocker and up the trans tunnel than it looked from inside, so give yourself extra margin. Where you hit good metal, that's where your patch goes.

For the patch, 18 or 20 gauge sheet works. I like making paper templates of the cutout first because the floor has some odd contours around the seat bracket and tunnel. Lap the patch over the good metal about an inch and plug weld through it, or butt it up and do stitch welds with small gaps so you don't blow through. Flux core on thin sheet is spattery and likes to burn holes, so practice on scrap of the same thickness until you can lay a bead without eating through. Grind the welds, then seam sealer over everythign.

One thing on the seat mount specifically: if the mounting holes themselves are in the rot, you'll want to weld in a thicker reinforcement plate there, not just floor metal, so the bolts have somthing to bite into. Before you weld, pull anything flammable out from under the Jeep, stuff a rag in the exhaust if you've got one nearby, and keep a fire extinguisher within reach. Also stick your head under there and poke at the rockers and frame rails while you're at it. Floor rot on these usally travels with friends.

Done a few of these going back to my old squarebody days. One trick from before everybody had a welder in the garage: drill the patch panel with a row of holes and use sheet metal screws to hold it in place while you tack it down a bit at a time. Keeps the panel from warping on you and pulling out of shape when it heats up. The flux core will blow through less too if you let each tack cool before hitting the next spot. Also check the wiring harness under the carpet before you close it up, those TJ's like to chew insulation right there where the moisture sits.

Check your body mount under the seat area before you even start cutting. I did a floor in my old F150 and the rot looked contained inside the cab until I popped the seat out and found the mounting bracket sitting on a body mount that was half gone. That turned a weekend job into a three weekend job real fast.

For steel, skip the auto parts store and hit a local fab shop or steel yard offcut bin. I grabbed a 2x3 foot blank of 20 gauge for cheap and had plenty left over for the reinforcement plate around the bolt holes. Also brush paint the back side of your patch before you weld it in, beacuse once it's in there you'll never reach it again.

Did a floor in my XJ a few summers back and my one regret was trying to work around the seat still bolted in. Pull the seat out first even though it feels like extra work. You'll need it out to reinforce the mount anyway, and having the whole area open makes cutting and fitting the patch about ten times easier. Also throw a moving blanket over the interior, flux core spatter travels and I still have little burn dots in my carpet as proof. After welding I brushed on some rust converter around the edges of the patch before seam sealing, seemed to stop the creeping orange around my repair.

This thread is making me nervous about my old Civic hatch, the trunk floor has the same blistered orange stuff going on and I kept telling myself it was surface rust. Sounds like it probably isn't. Question for you guys since I'm new here, when you're welding a floor patch do you do most of it from underneath lying on your back, or can you get away with reaching down through the cabin? I've only ever welded stuff sitting on a bench and the idea of running beads overhead on thin sheet sounds like a good way to rain hot metal on my own face. Also does the patch go on top so it's the new floor surface, or underneath?