Is it normal for a furnace to get red tagged one day and cleared the next?

A red tag is just that company's opinon written on a piece of paper. It's not a government order, nothing gets filed with the city, and there's no registry you're supposed to report to. You can toss it in the trash and legally nothing happens. I'd keep the invoice from the second company though, in case you ever sell the house and a buyer's inspector asks about it.

On the technical side, the second company did this the right way. Checking a heat exchanger properly means pulling the blower and getting a camera or scope inside. A tech with a flashlight and a mirror looking thru the burner opening sees rust and discoloration on a 16 year old unit all the time, and a lot of them call that a crack. Rust flakes and dye-penetrant marks on the seams get misread constantly on older furnaces.

And your instinct about the gas being left on is the part I'd trust most. If a tech genuinely believed there was a crack leaking combustion gases into your air stream, leaving the unit running and walking out the door woud be a hard thing to live with. The tag-and-quote pattern on an old furnace is a known move.

If the doubt keeps eating at you, one more option that doesn't cost much: some gas utilities will do a safety check themselves if you call and ask. They have zero incentive to sell you a furnace. Or hire a third company that does repair work rather than replacement, since they make money fixing things, not selling new ones.

Me personally, I'd run it, keep the second company's paperwork, and never let the first outfit back in the house.

Something similar happened to my dad's old furnace a few years back. Tech wrote it up, quoted a full swap, dad hired a guy he knew who did side repairs and the guy found nothing but a rusted burner orifice. Ran fine for three more winters before they sold the place. One thing I did at our own house after all that drama: I put a carbon monoxide detector near the bedrooms. Cheap peace of mind either way. If that furnace ever actually does crack, we'd know long before anyone could tag anything.

This stuff confuses me too. When our old heater got a bad report a while back, I asked the guy to actually show me the problem through the little window thing where the flames are, and he couldn't really point to anything that looked broken to my untrained eyes. That kind of settled it for me. Did your second company put their findings in writing on paper, or did they just tell your husband out loud? Feels like having something official-looking from them would help if the first company ever gives you trouble about it.

Update: we finally pulled the tag off. That first reply pretty much settled it for me, that it's just one company's opinion and nothing gets filed anywhere. I kept the second company's invoice with the furnace papers in case we ever sell. Also had our regular guy look at it during the annual service and he saw nothing wrong. Running fine.