In my story, all the fauna in the planet (basically Earth) has been instantly killed, leaving only flora and fungi. How long will it take before the effects start to show, and how long until all the CO2 disappears, (which would freeze up the planet)?
What you're describing may well not happen.
The assumption here is that ALL CO2 in the atmosphere is caused by animal respiration and that's simply not the case. One additional source of CO2 that hasn't been considered is plants, for instance. They produce CO2 when they respire and use the O2 and Carbohydrates they produce through photosynthesis to produce their own life energy. Granted, they generate less than they produce, but it's a factor.
Secondly, I'm going to assume that by 'animals' you're not including the microbial life in the soil that breaks down dead organic matter so that plants can then absorb it through their roots for nutrition. That also respires, producing CO2.
Next, you have volcanoes. They produce a massive amount of CO2 and it's generally believed in the scientific community that the end of the 'snowball earth' era was caused by a fresh outbreak of volcanic activity around the earth.
As such, that's the variable that would need to be accounted for in order to provide more precise estimates as to whether or not the earth would indeed enter another snowball earth epoch. Certainly animal respiration would reduce CO2 production, and getting rid of humans and industrial activity would also reduce it, but you've still got all the CO2 in the oceans to consider (Ocean-Atmosphere exchanges) and the impact of volcanoes to factor in to your equation.
Take a look at this XKCD temperature timeline which shows over time how temperatures have changed, and then compare that against any estimates of animal biomass that you can find in literature, and I'm pretty sure you'll find that the correlation is not particularly tight.
Bottom line is that the affect of removing animals could easily be countered by fluctuations (periodic increases in) volcanic activity before you see any appreciable effects in the environment, in my estimation.
Billions of years
A variety of plants will go extinct immediately, of course, in particular, the currently dominant flowering plants will lose their evolutionary advantage of having evolved alongside the animals that pollinate them. But there are plenty of primitive and otherwise plants that have never in their evolutionary history had any need for animal-driven pollination, such as conifers, cycads, and ferns.
With plenty of plants, there will be plenty of food for the fungus to live on, so life will continue, and thrive. In another hundred million years, some other branch of life will colonize the niches vacated by the now-absent animals, and we will have swimming fish-like foram protists or walking and flying fungus or something.
In any case, life will continue quite happily until the sun bakes all the water off the Earth in some billion years, and that is plenty of time for some fungus civilization to raise up and surpass our intellectual achievements.
If all the fauna on earth has been killed (vertebrates and invertebrates), there are no pollinators left. Which would kill off many plants. (That is, make them unable to reproduce, so once they die, there won't be any more of them.) The plants that are wind-pollinated will survive.
There will still be CO2: from fungi, from zooplankton. Plants also produce CO2. The earth would survive indefinitely. New species will evolve to fill up the emptied ecological niches.