How would this experiment's different particle types produce atmospheric reflectivity?
foreword: Particles and Droplets

Commonly, atmospheric particles can be considered as wet particles. E.g. when we talk about sulfate particles ($SO_4^{2-}$), these particles are actually droplets with dissolved sulfuric acid (or with a dissolved sulfate-cation compound).

Mechanism involving $SO_2$

$SO_2$ is released as gas and it reacts to $SO_4^{2-}$ (sulfate). Basically, there are different reaction pathways for the formation of sulfate from $SO_2$:


$SO_{2,g}$ --(gas phase reaction)--> $H_2SO_{4,g}$ --($H_2O$ accumulates)--> "sulfate particle"
$SO_{2,(g)}$ --(goes into solution)--> $SO_{2,solution}$ --(formation)--> $H_2SO_{4,solution}$ (deprotonizes directly)


Commonly, the second pathway is dominant. The relevant aspect is that it leads to the formation of ultra fine sulfate particles (sulfate droplets). 

Sulfate particles are cloud condensation nuclei. That means that they support the formation of clouds. There are satellite pictures available, in which you can identify shipping lanes by lines of clouds. These clouds were formed because ships emit large amounts of $SO_2$ (at least outside of sulfur emission control areas). 

Most clouds above the ocean reflect more solar radiation than the sea surface. In general, clouds above the ocean are considered as "good clouds" with respect to cooling the Earth. Since a large proportion of the Earth is covered by sea surface, one might wonder to generate clouds above it - e.g. by releasing $SO_2$. However, as far as I know, this process is relevant in the troposphere.

Releasing $SO_2$ into the stratosphere (as proposed in the linked article) is not meant to support cloud formation (correct me, if I am wrong). Nevertheless, the released $SO_2$ leads to ultra-fine sulfate particles. These sulfate particles themselves effectively scatter solar radiation and, thus, prevent it from reaching the Earth's surface. There is also a Wikipedia article available here.

Unfortunately, sulfate indirectly degrades the ozone layer: sulfate leads to the release of radicals that (and their successors) catalytically destroy ozone. See Keith et al. (2016) for a more detailed overview.

Why $CaCO_3$?

The basic idea is to add salts of alkali metals that neutralize sulfuric acid and nitric acid (generated in the presence of sulfuric acid) when we emit $SO_2$. This will reduce the negative impact on the ozone layer. $CaCO_3$ is such a salt (Keith et al., 2016). 

Why $Al_2O_3$?

Foreword: I am not expert on this and just read some articles on this topic.

According to Weisenstein et al. (2015), fine aluminum particles (as well as diamonds) are an alternative to sulfate particles. According to them, aluminum is favourable. However, as studies on emissions from missiles and space crafts indicate, $Al_2O_3$ also has negative impacts on the ozone layer (SomeReport; DoD, 2004): $Cl$ radicals are released that degrade ozone. $CaCO_3$ also counteracts here, too.