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).

I watched it, but it doesn't say anything about where they get the water from.

Yes it takes a couple of days.

No. It takes a couple of days if you would have algae plants at the exhaust of every CO₂-producing factory or vehicle. The algae plant you have there is especially effective because it gets a very high concentration of carbon dioxide. → Algae is pretty good at reducing emissions, but it cannot reduce the global levels within a few days.

Also you forgot one thing in your calculation: "What happens to the biomass?"
As mentioned in the video, most of it gets converted back to CO₂(mostly as fuel for vehicles and animals). So in the end the global levels will only insignificantly drop.

Additionally we have been seeing the greening of deserts over the last 30 years

As mentioned in the article: "Ongoing research is required if we are to fully comprehend the potential extent and severity of such secondary effects."
So it is unknown if those secondary effects are significant enough to reduce the slope of the current CO₂-rise.

And if you take a look at the provided satellite image:

You can notice there are a lot of areas that are yellow/red meaning the vegetation there actually dropped which opposes the positive effect in dessert areas.