You say you're in South Carolina which is 30% coal generation, not "mostly", and which is set to plummet rapidly over the next few years as several of the increasingly uneconomical coal plants are shut down.

The carbon tax should fund renewable energy sources.

...lots of things SHOULD be true, but are not in practice.

Emission reductions and or efficiency improvements are also possible outcomes if the tax is higher than the cost of these changes.

Only if you can finance/afford it. If not, it could even delay change as it reduces available funds.

The target is really industry. If no one has to pay the cost of emissions no one has any incentive to change.

Look to the oil crisis of the 70s for examples - it was a bad time but the cost of fuel spurred a surge in sale of small cars because fuel efficiency finally mattered. On another front it spurred the cycling culture of the Netherlands - they didn’t take up cycling and build infrastructure to support it out of altruism, they did it because their fuel supply was nearly cut off entirely.

Innovation only happens when there is a reason to innovate. If carbon emissions don’t cost the emitter anything then there’s no reason to invest in ways to emit less.

If the costs can be passed on, then there is potentially no incentive to innovate, either. It might also still be more cost efficient to innovate less to reduce costs than to actually innovate to reduce costs. Predicting where these things go is quite tricky.

But if the costs can be passed on then that’s a means for emitters to differentiate on cost. If manufacturer A emits more and passes the tax onto the consumer, manufacturer B can undercut on them on price if they’re more efficient.

Right now there’s minimal financial benefit to being more carbon efficient. If anything it’s disincentivised because efficiency is itself costly, so it’s cheaper to just emit.

But only if undercutting isn't too expensive/pays back fast enough. The issue is that from a revenue perspective things always look simple, from a profit perspective it gets more complicated, especially if things have switching costs/limited fungibility, are oligopolies and other structured markets. It might work like you outline but I would not be surprised if there are a lot of unexpected or undesirable results, too.

Impose a law that forbids the constructions of new fossil fuel power plants, and any existing plants must be decommissioned when their current planned operational life-time has expired.

Either the market will start to invest in non-fossil fueled alternatives when demand exceed supply, or people will elect governments that step in and invest in non-fossil fueled alternatives. Either way, the coal, oil and gas plant can not continue to be part of the European energy grid.

And then there’s the energy wasted in transmission.