To answer your questions here:

> Sure, but what about sea organisms turning into limestone and trapping carbon in the ground?

Too slow, sadly, and running into problems due to ocean acidification.

> What about cloud cover changing the rate of heat absorbed from the sun?

This is included in the models and one of the largest sources of error bars in the result. Clouds are complicated. We're working on it. (To deal with it one can put the highest and lowest probable value into the model and see what the difference is.)

> What about the plants that created the fossil fuels in the first place?

They lived in a different climate. What about them?

> How much time does it take for the effect to appear?

Which effect exactly are you talking about? Does appear mean measurable or clearly visible to humans? Depending on what you mean it is either very fast, just years, or takes some time, hundreds of years.
The ice ages seem to end rather abruptly (hundreds of years, maybe even less! , nothing on geological timescales), but we don't know why yet. The knowledge might not help with the current problem either. This rate, method and numeric range of change is unprecedented in the historic record. (400ppm of CO2 and above like we have now can only be found in the old atmospheres when the climate was very different.)
We can easily calculate how much additional energy is entering the Earth system, and that energy has to go somewhere (right now mainly the ocean). System reactions are less certain and can be slow.

> we'll probably be able to reactively adapt through technology

Maybe, if we hurry. This is like technical debt. Any CO2 we add now we have to either get out again, at high energy costs, or get used to the new situation. If you check the numbers it is a lot cheaper to fix it now than to keep going and then adapt/undo.

> Do we have some easy new theory they can apply? Do we have some new data they can plug into their simple formulas to make those predictions?

This has become something you need to do as your full-time job or time-consuming hobby to do. There is just so much going on. There is a lot more data (new methods of finding new data (eg satellites!), longer data records), and a lot more people (not just 2), computers! (you do not want to run all these calculations by hand), and we're all building on each other. The computers allow us to run more and more complex models, increase the resolution, include more effects. The input parameters and states are becoming clearer (we have daily measurements all over the globe now, back then they had what, 1 or 2 research sailing boats that came by every 20 years and didn't even measure much?). We have detailed knowledge about how much ice there is now, for example, and about the cloud cover every day, about storms and about ocean currents, wind patterns, natural emissions, atmospheric chemistry reactions (such as Ozone catalysation which led to the ozone hole, which does have a climate effect, just not very important). Any of these and more has specialized people studying it full-time. How can you keep up as a single person? You probably cannot, not in the detail required by the critical people.

It is interesting that this science has so many people trying to argue it away. Can they predict a moon eclipse accurately themselves? They still do not doubt the prediction. Can they predict the existence of the Higgs Boson, the effects of general relativity on the signal of GPS satellites, the exact path a solar system probe will take? There are so many things science predicts, but not a lot of people try to argue.

It's probably because everyone is asked to change, and fast, or be doomed. Maybe something about the communication went terribly wrong at some point and that is why we will have massive problems. (Interestingly, the ozone hole is a wonderful example of quick and decisive international collaboration in the face of a problem. Did people argue against the science then? The problem was simpler and smaller, but still of a similar nature.)