Where did all that CO₂ come from and where did it go to?

Earth is in homeostasis but does have periodic cycles from external sources which kicks it out.

Consider what happens as the sun is on the downward side of the cycle. With less energy earth starts to cool. As water freezes it sequesters CO₂. Anaerobic actions stimulate the production of Hydrogen Sulfide. This in turn is a natural fertilizer which stimulates more plant growth. Plants die and sequester CO₂ in the soil forming peat bogs. All of this takes CO₂ from the atmosphere although I am sure there are other mechanisms. Now consider what happens on the other side of the cycle.

When the amount of solar energy is at its lowest (during the glaciations), CO₂ is at its lowest. Once the sun starts going up in its cycle glaciers melt and a larger amount of area starts to darken.

The loss of polar regions might explain why the rise in temperature is steeper than the fall. While it takes tens of thousands of years for the earth to develop the full extend of the glaciers it only takes a few thousand years to go the other direction. As the albedo falls sequestered CO₂ becomes released from the environment.

Source
As water temperatures increase its CO₂ solubility falls. This produces 42.84% of natural carbon dioxide emissions.

Another aspect of the glaciers receding is that more and more organisms (including animals) start to respire. It isn't just cow farts affecting the planets but any organism which breaks down plant material contributes to this.

Another aspect of changed solar activity is an change in volcanic activity. As more energy is pumped into the system an increase in earthquakes and volcanoes occur.
While almost insignificant in relation to the total quantity of CO₂ it is additive to the levels.

The problem is water vapor will only reflect sunlight when it re-condenses in the atmosphere. This process requires lower temperatures, so I wouldn't be to sure about water vapor regulating everything.

"Water vapor accounts for 60-70% of the greenhouse effect while CO2 accounts for 25%". Water vapour is a dual edged sword. While it increases the storage of heat, it also has the ability to reflect heat once it reaches a particular concentration. The hotter the temperatures the more water vapour thus increasingly thick clouds. Thick clouds have an albedo of 75%. Additionally water vapour also sequesters CO₂. It is a feedback loop which self corrects in the long run.

This also comes into play with the question of sufficient water. Algae goes better in acidic water than alkaline water. Increased CO₂ encourages growth of algae as a natural fertilizer but by also increasing the amount of other nutrients which are bound up in other forms. Additionally we have been seeing the greening of deserts over the last 30 years

Where do you get all that fresh water?

This video is 7 minutes but I would like you to watch this from the perspective that humans need to work with the earth so it can do what it does best. The temperature in the area that this project has reached a high of about 109 f (43 c).

This video should give a reasonable overview of algae production. It also addresses your comment about "And as I said CO₂ needs a long time to be reabsorbed and formed to new oil/coal. " Yes it takes a couple of days.